Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

Approximation Theory in- Tensor Product Spaces

S pringer-Verlag Berlin Heidel berg New York Tokyo

Authors

William Allan Light Mathematics Department, University of Lancaster Bailrigg, Lancaster LA1 4YL, England

EllJott Ward Cheney Mathematics Department, University of Texas Austin, Texas ?8712,USA

Mathematics Subject Classification (1980): Primary: 41 A63, 41 A65 Secondary: 41-02, 41A30, 41 A45, 41A50

ISBN 3-540-16057-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16057-4 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

D E D I C A T I O N

This work is dedicated to the memory of

Robert Schatten

(1911- 1977)

who did much of the pioneering work in

the theory of tensor products of Banach spaces.

P R E F A C E

In the past two decades, a new branch of approximation theory has emerged; it con-

cerns the approximation of multivariate functions by combinations of univariate ones. The

setting for these approximation problems is often a Banach space which is the tensor prod-

uct of two or more simpler spaces. Approximations are usually sought in subspaces which

are themselves tensor products. While these are infinite dimensional, they may share

some of the characteristics of finite-dimensional subspaces. The usual questions from clas-

sical approximation theory can be posed for these approximating subspaces, such as (i)

Do best approximations exist? (ii) Are best approximations unique? (iii) How are best

approximations characterized? (iv) What algorithms can be devised for computing best

approximations? (v) Do there exist simple procedures which provide ~good = approxima-

tions, in contrast to ~best = approximations? (vi) What are the projections of least norm

on these subspaces? and (vii) what are the projection constants of these subspaces?

This volume surveys only a part of this growing field of research. Its purpose is

twofold: first, to provide a coherent account of some recent results; and second, to give

an exposition of the subject for those not already familiar with it. We cater for the needs

of this latter category of reader by adopting a deliberately slow pace and by including

virtually all d e t a ~ in the proofs. We hope that the book will be useful to students of

approximation theory in courses and seminars.

Expert readers may wish to omit a reading of the first chapter, which gives an intro-

duction to the tensor product theory of Banach spaces. The material on approximation

theory occupies the next eight chapters. Results needed in proofs but perhaps not familiar

to every reader are collected in two appendices (Chapters 10 and 11). Finally, there are

historical notes and a large collection of references, some of which are only peripheral to

our theme.

Notation and conventions are standard throughout, and we often do not stop to define

notation which we expect to be familiar. However, a table of notation has been placed

just before the index.

We are glad to be able to thank a number of colleagues for pleasant collaboration over

the years on matters relating to these notes: Carlo Franchetti (Florence), Manfred von

Golitschek (Wfirsburg), Julie Halton (Lancaster), Sue Holland (Lancaster), John Respess

(Austin), and Lin Sulley (Lancaster and Ipswich).

During the preparation of the manuscript, the second author was supported by grants

from the University of Texas and the Science and Engineering Research Council of Great

Britain. For these grants, and for the hospitaliW of the University of Lancaster, he is

deeply grateful.

We are very much indebted to Ms. Jan Duffy of the University of Texas Mathematics

Department, who undertook the arduous task of rendering our manuscript into a computer

file for processing by the TEX typesetting system. The pleasing appearance which (we

think) the book possesses is due entirely to the skill and good judgement of Ms. Duffy.

W.A. Light E.W. Cheney

Lancaster, July 1985

vi

C O N T E N T S

1. AN INTRODUCTION TO TENSOR PRODUCTS

2. PROXIMINALITY

3. THE ALTERNATING ALGORITHM

4. CENTRAL PROXIMITY MAPS

5. THE DILIBERTO-STRAUS ALGORITHM IN C(S x T)

6. THE ALGORITHM OF VON GOLITSCHEK

7. THE L1-VERSION OF THE DILIBERTO-STRAUS ALGORITHM

8. ESTIMATES OF P R O J E C T I O N CONSTANTS

9. MINIMAL PROJECTIONS

10. A P P E N D I X ON THE BOCHNER INTEGRAL

11. A P P E N D I X ON MISCELLANEOUS RESULTS ON BANACH SPACES

NOTES AND REMARKS

BIOGRAPHICAL SKETCH OF ROBERT SCHATTEN

REFERENCES

INDEX OF NOTATION

INDEX

1

35

48

56

60

67

75

91

103

113

126

134

138

141

153

156

C H A P T E R 1

A N I N T R O D U C T I O N T O T E N S O R P R O D U C T S P A C E S

The purpose of this chapter is to introduce some of the basic theory of the tensor

product of two Banach spaces. All of this mater ia l can be found in other sources, but the

t rea tment here is part icularly designed to meet the needs of the subsequent chapters. As

mentioned in the preface, we have tried to expound the subject in a way tha t leaves very

few arguments for the reader to supply.

There are two sources of information on this topic which we should mention at the

outset. One is Schat ten 's monograph [154], which gives a very careful t rea tment of the

foundations of the subject. The other is the survey of vector measures by Diestel and Uhl

[55]. Chapter 8 of [55] provides a brief introduction to tensor products of Banach spaces

and covers many recent results in this area.

Let X and Y be Banach spaces, and denote their duals by X* and Y*, respectively.

We shall construct formal expressions ~ ® y~ where x~ E X, y~ E Y and n E 1N. We ~ i - - - - 1 Xl

will regard such an expression as defining an operator A : X* -* Y, given by

r$

A¢ = ~ ¢ ( x , ) y , (¢ C X ' ) . i----1

Amongst all these formal expressions we introduce the relation

~ ~ x i ~ Yi "" ai ~ bi i m l i = l

if bo th expressions define the same operator from X* to Y. This is clearly an equivalence

relation on the set of all such formal expressions. We shall henceforward be interested only

in the equivalence classes of this relation, and will denote the set of all such equivalence

classes by X ® Y. We shall abuse notat ion in the usual way by referring to the expression n ~ = 1 x~ ® Yi as a member of X ® Y when we intend to refer to the equivalence class of

expressions containing ~ " = 1 x~ ® y~. For any c~ E ~ we define multiples of (the equiva-

lence class of expressions containing) ~ n i=1 xl ® y~ by (the equivalence class of expressions

containing) ~"--1 ax~ ® y~. Similarly, we define addition by

xi N Yi -5 xi N yi = xi N Yi. i = 1 i - - - -n+ l i = l

All algebraic identities in X ® Y are based upon the interpretation of expressions as linear

operators. Thus one easily verifies such identities as

z ® ( u + v ) = z ® u + x ® v

a x ® y = x ® a y

z ® 0 = 0 ® 0 .

Two observations are perhaps helpful at this juncture. Firstly, the symbol 4+, is being

used only as a separator in our formal expression ~ - - 1 x~ ® y~ and in our definition of the

addition of two such expressions. Secondly, the construction thus far makes no use of the

topological structure of X or Y. Thus we could have begun with linear spaces and algebraic

duals rather than Banach spaces. If we begin with Banach spaces and identify n E i = I xi®Yi

with an operator A mapping the algebraic dual of X into Y then the equivalence classes

that make up X ® Y remain the same.

It is clear that scalar multiplication of ~ i x~ ® y~ by ~ is equivalent to multiplying

the associated operator A by (x, and that the addition of this expression to ~ 5 = i aj ® bj

is equivalent to adding A to the operator B associated with this last expression. With

these definitions X ® Y forms a linear space, called the a l g e b r a i c t e n s o r p r o d u c t .

r$ 1.1 L E M M A . Every expression ~i=i xi @ Yi is equivalent to ei ther 0 ® 0 or to an

expression Eim= l ai ® bl where { ai, . . . , a m ) and { h i , . . . , bin} are linearly independen t sets.

n--i n P R O O F . Suppose that for instance x,~ : ~ j = i a j x j . Then ~ i = i xi ® y~ defines the

operator A : X* ---* Y where, for ¢ E X*,

A ¢ = ¢(x,)y~ = ~ ¢(x , )y i + ¢ ( x , ) y , i = 1 i = 1

n - - 1 n - - 1

n--i n--i

: + j¢CxAv. i = 1 3 = 1

n - - 1

= + v . ) .

i----1

Hence ~ ,~- 1 ~i=i x~ ® Yi has the representation ~=i a~ ® b~ where a~ = x~ and bi =

Yi + ~i Y,~. We may repeat this process until we arrive at either a representation in which

{ a i , . . . , am} and {b i , . . . , bin} are linear|y independent or one of the representations x ® 0

or 0 ® y, which are both equivalent to 0 ® 0. •

The space X ® Y is generated by elements of the form x ® y, which are called d y a d s .

This observation is often used to simplify linear arguments.

From the preceding discussions it is clear tha t X ® Y may be regarded as a subspace

of the space of continuous linear operators of finite rank f rom X* into Y. Usually this

subspace is a proper one. However, in the case when X is reflexive every continuous finite

rank linear opera tor from X* into Y can be identified with an expression ~-]~$=1 xl ® Yi

as follows. Suppose A is such an opera tor with range determined by linearly independent

b t , . . . , bn. Then for ¢ E X*,

A ¢ = where e R . i = l

Standard arguments show tha t the ai are in X** and hence in X, whence a i (¢ ) = ¢(a~)

for suitable ai E X, 1 < i < n. Hence,

A ¢ = ¢ (a , )b ,

i = l

and A is associated with ~ i ~ t ai ® hi.

It is possible to construct various norms on X ® Y using the norms in X and Y. The

most obvious way to introduce a norm which is independent of the representat ion of the • l l

equivalence class is to assign to ~ i = 1 xi ® Yi the norm it receives when regarded as an

opera tor from X* to Y; viz.,

xi ® yi = sup * 1

Notice tha t for a dyad x ® y we have

i = 1

A ( x ® y ) = s u p { l l ¢ ( x ) y l t : ¢ e x * , Hell = 1}

= sup{]¢(x)IIly]]: ¢ e x * , ]]¢ll= 1}

= IIxll Ilytl.

Such a norm on X ® Y, for which the norm of a dyad equals the product of the norms

of its two components, is termed a c r o s s n o r m . Given any two Banach spaces X and Y

there is a rich supply of crossnorms on X ® Y.

We can form a second linear space from X and Y by considering X* ® Y*. This

consists of all expressions

f i ¢ ~ ® ¢i where ¢~ E X* and ¢i E Y*. i = 1

In addition to other interpretations such an expression may be considered as a linear form

on X ® Y by defining

¢, ® ¢, x~ ® y~ = ~ ¢,(~J)¢,(~) . i : l "---- i = l 3'----1

One verifies easily that this definition is proper; tha t is, it is invariant over the equivalence

classes. Note tha t it would have sufficed to give the above definition for a pair of dyads:

(¢ ® ¢)(~ ® y) = ¢(x)¢(y).

1.2 D E F I N I T I O N . Let a be a norm on X ® Y. We say that ~ is a crossnorm if,

for all x E X and y E Y~

-(~ ® y) = II~1t Ilyll.

We say that a is a reasonable norm if, for all ¢ E X* and ¢ @ Y*, the linear form ¢ ®

is bounded on ( X ® Y, a) and has norm equal to I1¢11 II¢11.

1.3 D E F I N I T I O N . Let a be a norm on X ® Y. De~ne ~* on X* ® Y* by the

equat ion

a* ¢ , ® ¢ , ----sup ¢ , ( x j ) ¢ , ( y j ) : a ~ x y ® y j = 1 . i----1 i = 1 3'----1 3 ' = 1

We admi t the possibi l i ty that a* m a y take ÷oo as a value.

Observe that if a is reasonable then a* is the norm induced on X* ®Y* by considering

the lat ter as a linear subspace of (X ® Y, a)*. The norm a* is the a s s o c i a t e of a.

1.4 L E M M A . Let a be a norm on X ® Y satis fying

(i) a ( x ® y ) <_ I[xlt[[yN for all x e X and y e V ;

(ii) a * ( ¢ ® ¢ ) <[1¢[[[1¢11 for all C E X * and C E Y * .

Then ~ is a reasonable crossnorm.

P R O O F . If x E X , y E Y, C E X * , a n d C E Y * , t h e n

¢(~)¢(y) = (¢ ® ¢ ) ( z ® y) < ~*(¢ ® ¢)~(~ ® y) < ~ II¢l111¢11~(~ ® y)

- - L I1~11 I l y l l ~ * ( ¢ ® ¢).

In this inequality, take a supremum as ¢ and ¢ range over the unit cells of X* and

Y*, getting

IlxtJ Ilyll < a ( x ® y).

4

If we take a s u p r e m u m as x and y r ange over the uni t cells of X and II, we have

I1¢11 I1¢11 < ~*(¢ ~ ¢). a

1.5 L E M M A . S u p p o s e t ha t c~ is a reasonab le c r o s s n o r m on X ® Y. T h e n ~* is a

r ea s o n ab l e c r o s s n o r m on X * ® Y * .

P R O O F . Since c~ is a r ea sonab le no rm, a* is a c rossnorm. In o rde r to show t h a t c~* is

a r e a s onab l e no rm, t a k e / ~ E X** and u E Y**. By 1.4 we need on ly show t h a t

~** (" ® ") -< ll~ll II"ll.

By G o l d s t i n e ' s t heo rem [57, p. 424] there exis t nets

{~} c x, {y~} c y

such t h a t ~ and v are the weak* l imi t s of {x~} and {yz}, and

.11~11 < I1,11, Ily~ll < I1,'1t.

Let

= ~ ¢ ~ ® ¢~ ~ X* ® Y*. d = l

Then

By t a k i n g a l imi t in fl and "7 we get

leC~ ® ,')l -< ~*(e)11~ll II,'ll

or I (~®") (e) l _< ~*(e)ll~ll II,,ll.

This shows that ~ * ' ( ~ ,,) _< I1~11 I1"11. n

R e t u r n i n g now to our only e x a m p l e thus far of a c rossnorm, we refer to A as the leas t

of the r e a sonab l e c rossnorms by v i r t ue of the fol lowing l emma .

1 .6 L E M M A . The n o r m A is a reasonab le c r o s s n o r m on X ® Y. Moreover , f f a is

a n y reasonabIe c r o s s n o r m on X ® Y then

A(z) <c~(z) for all z e X ® ] 1 .

P R O O F . In our p r e l i m i n a r y d iscuss ion of A we verif ied the c ros sno rm p rope r ty . We

h a v e

( £ ) } (1) A x , ® y , - - s u p x , ) y , : ¢ e X * , ll¢ll-= 1 .

I 1 i - ~ l

5

If n o w C E X * , C E Y * and z is any element of X ® Y then by Eq. (1)

--- 11¢1t II¢llA(z).

Thus A*(¢ ® ¢) _< II¢1111¢11 and so by 1.4 A is a reasonable crossnorm.

Now let a be a reasonable crossnorm on X ® Y. Let z E X ® Y, ¢ E X* and ¢ E Y*.

Then

I (¢® ¢)(z)l ~ ,~*(¢ ® ¢)~(z) -- I1¢11 II¢lI~(z).

Taking a supremum for ¢, ¢ belonging to the unit cells of X* and Y* and using Eq. (1)

gives X(z) _< a(z). •

It is also possible to define the greatest crossnorm. We do this next and then proceed

to investigate each of these norms in somewhat greater detail.

1.7 D E F I N I T I O N . For any z E X ® Y we define

"~(z} = i n f { ~ llxil[ I,Yill : xi ~ X, y~ ~ Y, z = ~ x, ® y, } . i = 1 i = 1

1.8 L E M M A . The norm ~ is a reasonable crossnorm on X ® Y. Furthermore, i f

is any crossnorm on X ® Y then 3(z) >_ aCz) for all z E S ® Y.

P R O O F . The first issue at stake here is whether 7 is a norm at all. We check first that

if z ~ 0 then 7(z) > 0. Let

z : E x i ~ y i . i = 1

If z is nonzero then the operator Lz defined by z is not the ~ero operator from X* into Y.

Hence there exists a ¢ ~ X* such that 11¢11 = 1 and L~¢ ~ 0. Thus

~ ¢ ( z , ) y , " o < IIn~¢ll ~ -< ~ I¢(~,)1 IlY, II ~ IIx, II IlY, ll. / = 1 / = 1 i = l

Since this holds for a// representations of z we have 7(z) > 0. The definition of 7(z) is

clearly independent of the representation of z E X ® Y and the remaining properties of a

norm are easily verified.

To establish that 7 is a crossnorm, suppose that x ® y and n ® Yi two ~ i = 1 xi are

equivalent expressions for z E X ® ]I. Then for all ¢ E X* we have

¢(x)y = ~ ¢(x,)y, . i = l

Now choose ¢ e X* so that licit = i and ¢(x) = tl~]l. Then

It~11 Ilylt = It¢(~)yII = ~ ¢ ( ~ , ) y , < ~ tl~,ll Ily, ll. i = l i = l

Hence "r(z) = I1~1t Ilyll.

To show that "r is a reasonable norm~ take

z = ~ x ~ ®Yi E X ® Y, i = 1

a n d C E X * , C E Y * . T h e n

I(¢ ~ ¢)(z)t -< ~ 1¢(~,)I I¢(Y,)I i = l

<- 11¢tl I1¢11 ~ I1~,11 tly, tl. i = 1

Since this bound holds for all representations of z, we have

I(¢ ® ¢)(z)l _< 11¢11 li¢ll~(z),

so that "r*(¢ ® ¢) < II¢11 II¢ti. An application of 1.4 completes the argument.

Finally, we show that 7 is the greatest crossnorm. Let a denote any crossnorm on

X ® Y and let

Then

z = E x i ® y ~ E X ® Y. i..= l

~(z) = ~ ~, e y, <_ ~(~, ® y,) = t1~,tI Ity, ll- - - i : l "= i = I

Taking an infimum over all representations of z, we get a(z) <_ 2(z) . •

1.9 L E M M A . Le$ X and Y be Banach spaces and suppose that a and fl are two

norms on X ® Y with a >_ 8. Then a* <_ fl*.

P R O O F . Let 0 E X* ® Y* be represented by

i = l

Then

~*(e) = sup {

_> sup {

E ( ¢ i ® ¢ i ) ( z ) : z E X ® Y , ~(z) < 1 i = 1

i = 1

= ~ * ( ~ ) . •

1.10 L E M M A . Let X and Y be Banach spaces and let a be a crossnorm on X ® Y.

Then a* is a crossnorm on X* ® Y* i f and only i f c~ > A.

P R O O F . Suppose tha t a > A. Then 1.9 gives a* _~ A* and so

~*(¢ ® ¢) < A*(¢ ® ¢) -- I1¢11 II¢ll for all ¢ E X*, ¢ E Y*.

Also

II¢tl II¢II = sup{I¢(~)t I¢(y)l : • • x , y E Y, !!~II = llyll = 1}

_< sup{~*(¢ ® ¢ ) ~ ( ~ ~ V): ~ e X, V • Y, LI~II = Ilvll = 1}

= ~*(¢ ® ¢).

Conversely, if a* is a crossnorm on X* ® Y* then a is a reasonable crossnorm. Hence by

1.6 we have a _> A. •

1.11 D E F I N I T I O N . Let X and Y be Banach spaces. A crossnorm a on X ® Y is

said to be uniform f f f o r every pa / r oE operators, A E ~ (X , X ) and B E r (Y , Y ) , we have

a Axi ® Byi <_ NAN IlBIla x~ ® y~ for all xi E X, Yi e ii. i = 1 i = 1

Later in this chapter (1.29) the tensor p roduc t A ® B of two opera tors will be defined. Then

it will become clear tha t the uni formity of the crossnorm a is equivalent to the assertion

tha t for all A and B, the n o r m of A ® B, as an opera to r on (X ® Y, a) , is HAll I[BII-

1 .12 L E M M A . Both A and 7 are uniform crossnorms on X ® Y.

P R O O F . Let A E £ ( X , X ) , B E £ ( Y , Y ) , C E X * , C E Y * and ]]¢H = ] [¢] [= 1. Then

(° ) ° %b E ¢(Ax')BY~ = E ¢ ( A x ' ) ¢ ( B Y i )

i----1 i = 1

= (A*¢ ® B * ¢ ) x~ ® y~

) i = 1

= I IA*¢] t I IB*¢I I ,X z, ® y~ i = 1

-< IJAII ]IB[I~X z, ® y~ . - - i = 1

If we take a s u p r e m u m on ¢ we get

~ ¢(A~,)Bv, < IIAlll]BIl~ ~, ® V, i : l i = l

If we take a s u p r e m u m on ¢ we get

(~= Axi ® <_ 1

T h e n

For "~ we take

z= f i x ~ ® y ~ E X ® Y . i = l

) i : l

"~ ~- 'Ax, ® By, <_ IIAx, H HBy, II < llAII IIBN llx, ll lly, lI. i:i /=i /----I

By t a k i n g an in f imum over a l l r e p r e s e n t a t i o n s of z, we get

"~ E Axd ® By, ~_ IIAII IIBll'~(z). i = l

|

In general , the space X ® Y e q u i p p e d wi th a r easonab le c rossnorm is not comple te .

We deno te the comple t i on of X ® Y wi th respec t to c~ by X ®~ Y. One of the a t t r a c t i v e

fea tu res of t ensor p r o d u c t theory is t h a t f ami l i a r spaces can be rea l ised as tensor p r o d u c t s

of more e l e m e n t a r y spaces wi th respec t to a su i tab le c rossnorm. We prov ide some of these

r ea l i s a t ions next . W h e n X and Y are Banach spaces which are i some t r i ca l ly i somorph ic

to each o ther , we wr i t e s imp ly X -- Y.

If S is a c o m p a c t Hausdorf f space and Y is a Banach space, C(S, Y) denotes the

Banach space of all con t inuous m a p s f f rom S into Y wi th no rm defined by

1 .13 T H E O R E M .

Ilfll : sup IIf(~)ll, 8ES

For any compact Hausdorff space S and any Banach space Y,

c(s) ®~ Y = c(s, Y).

P R O O F . W i t h each e lement

z= E x i ® y i i:= l

in C(S) ® Y we assoc ia te an e lement F , in C(S, Y) by defining

F,(~) = ~ x,(~)y,. i = l

Observe tha t

sup C E Y *

11¢11=1

= sup sup ¢ s

: sup sup 8 ¢

= sup liE. 8

= IIF~[[.

i = 1

(~)11

Thus the l inear m a p z ~-~ Fz, after being extended by continuity, is an i sometry of

C ( S ) ®x Y into C(S, Y). It remains to be proved tha t the image of C(S) ® Y is dense in

C(S, Y). Let f @ C(S, Y) and let e > 0. Since f is cont inuous and S is compac t the set

K = f(S) is compac t and hence to ta l ly bounded. Therefore there exist points Y l , - - - , Yn

in K such tha t the open cells B(yi, e) cover K. By the theorem on par t i t ions of uni ty [148,

p. 41], there exist funct ions g l , . . - , g~ in C(K) such tha t

O< g~ <__ 1, g~(v) = 0 when v c g\B(yi ,e) , and f i g ~ = 1. i = 1

Define zi = gi o f . Then

x iEC(S) , O<xi_< 1 and ~ x i = 1 . i = 1

n Pu t z = ~-~i=1 x~ ® Yl. If s E S, then

/ = 1 i = 1

_ ~ x , ( s ) l l / ( ~ ) - Y, II < , , / = 1

because x i ( s ) = 0 w h e n I]f(s) - yell ~> c, T h i s proves t h a t I l l - F~II < c. m

1 .14 C O R O L L A R Y . Let S and T be compact Hausdorff spaces. Then

C(S) ®~ C(T) = C(S x T).

P R O O F . In the previous theorem we take Y = C(T) and so ob ta in

C(S) ®x C(T) = C(S, C(T)).

10

It is now elementary to make the identification C(S, C(T)) = C(SxT) . With f E C(SxT)

we associate

/ e C(S,C(T)) where /(s} = ] 's

a n d fs is a section of f defined by fs(t) = f(s,t). II

The next theorem refers to spaces of the type LI(5 ' ,Y) , consisting of Bochner inte-

grable functions defined on a measure space S and taking values in a Banach space ]I. In

Chapter 10, the theory of such spaces is outlined.

1.15 T H E O R E M . For any measure space S an,3 .any Banach space ii,

L~ (S) ®~ Y = L 1 (S, r).

P R O O F . With each element

z = ~ x , ® ~ , in L l ( s ) ® r i = 1

we associate a function F~ : S --, Y by defining

r,(s) = £ x,(~)y,. i = 1

Since each xi can be approximated in LI(S) to any given precision by simple functions,

the same is true of Fz. As for the norms we have

f f " IIF,~II = I I F . ( s ) l l d s = ~:,(.~1 _< I ~ , ( s l l l l Y , II i = 1 i = l

i----1 " i = 1

ds

This shows that Fz E Li(S,Y). (Refer to Chapter 10.) By taking an infimum over all

representations of the element z, we obtain l[Fz I1 <<- ~t(z). The linear mapping z ~-, Fz thus

has a continuous extension to L1 (S) ®u Y. In order to see that this map is an isometry,

we show that when F~ is a simple function, [[F~[[ = ~t(z). In this case, we can assume that

x ~ x j = 0 for i ~ j and £ x i = 1. i = l

Then

~,(s)y, d~ -- t~,(~)l I!y, Jlds -- ~,Lt Ily, Jl >- d z ) , i = 1 i=l i = 1

Since the simple functions of the form Fz are dense in Li(S,Y), this completes the

proof. II

Using the Fubini theorem and the ideas of 1.14 we have the following result:

11

1.16 C O R O L L A R Y . If S and T axe a-finite measure spaces, then

LI(S) ®~ LI(T) = LI(S × T).

Let a be a n o r m on X ® Y, and let G and H be subspaces of X and Y respectively.

Then G ® H is a subspa~=e of X ® Y which inherits the na tu ra l norm c, I G ® H. The

comple t ion of G ® H with this norm is G ® ~ l c ® g H. This is no th ing bu t the closure of

G ® H in X ®~ Y. To avoid cumbersome nota t ion, we write this closure as G ~ H. (The

exact mean ing of this no ta t ion will therefore depend on the context .)

For a no rm ct tha t can be defined on the tensor p roduc t of any pair of Banach spaces,

it is possible for G ~ H to differ f rom G ®~ H. For example, if a -- ~/ and z E G ® H,

then we have

( ' l l G ® H ) ( z ) = i n f Hx, H ]]y,]l : z = x , ® y , , x, e X , y, E Y i = l i = l

while

{5 5 ) - / ( z ) = i n f IIg, l l l lh~l I :z= g , ® h , , g, e a , h, e H . i = 1 i = 1

Therefore "~(z) >_ ( ' / I G ® H ) ( z ) for z E G ® H . In general, a strict inequali ty will hold here,

and G ~ H will be different f rom G ®; H. See [55, p. 227]. The next theorem addresses

this quest ion for the ;~-norm.

1 .17 T H E O R E M . Let U be a dosed subspace of the Banach space X and let Y be

a Banach space. Then U ®~ Y is a dosed subspace o f X ®~ Y.

P R O O F . Let us adop t t emporar i ly the no ta t ion Au®r and A x ® r for the A-norms of

an element z when though t of as a member of U ® Y or X ® Y respectively. It will suffice

to show tha t if z is in the l inear space U ® Y then

Y~ If z = ~ i = i ui ® Yi, then

)~v~r (z} = sup { : ¢ E u*, ll¢l] = 1}. i = l

By using the Hahn-Banach theorem to extend each ¢ e U* to a ¢ E X* wi th I1¢][ -- I1¢1[

we see tha t

(z) < x®Y (z)

By restr ict ing funct ionals in X* to act only on U we obta in the reverse inequality. []

12

1.18 L E M M A . Let X and Y be Banach spaces.

exist x,~ E X and y,~ E Y such that

oo

r t = l

Cii) I1~.11 -- Ily.II -~ 0, o o

(ii¢) w(~) -< ~ I1~.1111y=ll <- wCz) + ¢.

I f z C X ®~ Y and e > 0 then there

P R O O l e . Select zn c X ® Y such tha t

" , / ( z - z , ) < e / 2 "+9", for n = 1 , 2 , . . . .

F rom the definition of % we can write

k(1) k(1)

zl = ~ xd ® Yd where ~ I1~,1111y, II < ~ ( ~ ) + ~/4 < ~(z) + ~/2. i = 1 i = l

It is clear tha t here (and below) we can assume IIx,]] = Ily, ll- Now for each n,

~(~ .+~ - z . ) < ~ (~ .+~ - z) + ~(~ - ~ . ) < ~/2"+~ + ~/2"+~ < ~/2 "+1

Thus we can represent zn+l - z,~ as follows:

k ( n - t - 1) k (nA- 1)

z,~+l - z,~ = ~ x, ® y, with ~ Nx'[[ IlY'II < E/2~+1" i=k ( , )+ l i=k(,~)+l

The series ~ 1 x~ ® Yi is absolutely convergent in X ®-y Y since

~ ( ~ , ® y,) = il~,li Tiy, ll = ~ ll~,ii lly, il + i : l i : l i = l n = l i = k ( n ) - 4 - 1

oo

< ~(z) + ~/2 + ~ ~/2 "+1 = ~(~) + ~. r t ~ l

The series ~ i ~ 1 xi ® Yi is therefore convergent in X ®~ Y. Its l imit is z, since

.k(,~)

" i = l

= ~(~,~ - z / - ~ o.

It follows tha t oo oo

~(z) _< ~ ( x , ® y,) = ~ ,,x, JJ lly, il < ~(z) + ~. i = 1 i ~ 1

IIx, II Ily~ll

13

S i n c e Ilxill llY~ll ~ 0 a n d IIx, II = Ily~ll, w e m u s t h a v e IIx, II ~ 0 a n d IlY~II - ~ 0 •

1.19 D E F I N I T I O N . Let X and Y be Banach spaces, and let a be a norm on X ® Y.

Let A E T-(X,Y*). If the sup remum

sup Az,)(y~) : a xi ® yi = 1 i = 1 -- i=l

is ~.~te, we denote it by 11AII~. The s~t o~ aU operators having IIAII- < oo is denoted by

~ ° ( x , Y * ) .

In this definition the quest ion of the independence of the value of ~ i ~ (Axi)(y~) over

all possible expressions represent ing an element z E X ® Y does not arise. However it

becomes impor t an t in the next theorem and so we establish this fact now.

1 .20 L E M M A . Let X , Y and Z be Banach spaces, f f

~ z i ® y i ~ 0 ® 0 i = l

in X ® Y then r t

~ ( a ~ , ) ( u , ) = o i = 1

/'or all A E T-(X, T.(Y, Z)) .

P R O O F . By the definition of equivalence of expressions,

¢(u,)~, = o i=1

for all ~b E Y*. If { X l , . . . , x , } is l inearly independent , then ~b(yi) = 0 for 1 < i < n and

for all ~b E Y*. Hence Yi -- 0 for i = 1 , . . . , n and

r~

E ( A x , ) ( y , ) = O. i = 1

If {X l , . . . , xn} is linearly dependent then, say,

Zn ~ E )tixi" i = 1

As in the p roof of 1.1, we have

n n - - 1

~,®v,~ ~ x, ® (y, + ~ ,y . ) ~ 0 × 0 i = 1 i = 1

14

Also, by similar algebraic manipu la t ions

n n - - 1

~(A~,)(y,) = ~(Ax, ) (y , + ~,y.) i = 1 i = 1

If { x l , . . . , x,~-l} is l inearly independent , then as above we conclude tha t

Yi + ~iY,~ = 0 for 1 < i < n - 1 and tha t E ( A x l ) ( y i ) = O. i = 1

By cont inuing the above a rguments we either establish tha t

~(A~,)(y,) = 0 i = 1

or arrive eventual ly at a dyad u ® v such tha t

u ® v ~ x i ® y i N O × O , i = l

and

( A u ) ( , ) = ~(A~,)(y,) . i = 1

Since either u = 0 or v = O, we have

r$

~(Ax,)(~,) = o • i-=1

1.21 T H E O R E M . /YX and Y are Banach spaces, and if(~ is a crossnorm on X ® Y ,

then

( X ®, Y)* = £ , ( X , Y * ) .

P R O O F . Corresponding to an element 0 E (X ®a Y)*, define an element

by set t ing

Ao E £ ( X , Y*)

(A0~)Cy) = e(~ ® y).

It is clear tha t Aex is a l inear funct ional and tha t Ae is a linear operator . T h a t Aox E Y*

follows f rom the inequali ty

I(A0x)(Y)I -< II011a( x ~ Y) = II01111xll IlYlI.

15

In order to see tha t Ao E £~(X , Y*) we write

• ( A o • , , ) C y , = 0(~, ® y,) = 0 x, ~ y, "= i = 1 -- i = 1

By taking a supremum over all expressions of unit o~-norm we obtain IIA011~ = Ilell. Thus

the mapping 0 ~-* Ao is an isometry of (X ®~ Y)* into L~(X, Y*). In order to see that

this mapping is surjective, let B be an element of ~ ( X , Y*). Define 0 by

The functional 0 is well-defined by

since

i = l x i ~ =

Finally, we note that Ao = B. |

) Sc i----1

1.20. The linearity of 0 is clear, and ~ E (X ®~ Y)*

~ - ~ ( B x , ) ( y , ) < o, ~, ® u, I IB I I . . i = l ' i = l

1.22 C O R O L L A R Y . Let X and Y be Banach spaces. Then

(x e , Y)* = e(x , y*).

P R O O F . Using the preceding theorem we see that

(X®~ r*) = ~.~(x ,r*) .

Thus we need only show that £~(X, Y*) =: /?(X, Y*). If

A C ~ ( X , Y * ) and z = ~ x i ® y i , i = 1

then

~ ( A ~ , ) <_ I(A~,)(U,)I <_ ~ IIA:~,II IIY, II _< IIAII I I~, I I I IY, II. i = l i ~ l i ~ l i----1

Taking an infimum over all possible representat ions of z E X ® Y , and using 1.20 we obtain n

~ ( A x , ) C y , ) < I IAII '~(~).

Taking a supremum over z with "~(z) -- 1 gives IIAII~ -< IIAII. For the reverse inequal i ty

take z e X such that I1~11 = 1 and IIAxll -> I I A I I - ~- Choose y e V, IlYil = 1, such that

I (A~) (Y) I >- I I A ~ I I - ~. Then

IIAII -< IIAxll + ~ -< ICA~)(Y)I + 2~ _< IIAtI~ + 2~. •

Our next objective is to prove similar theorems involving ~ (the least of the reasonable

crossnorms). Some preliminary definitions and results are needed before proceeding to

these theorems.

16

1.23 D E F I N I T I O N . A Banach space X has the approximation property/ f /or each

compact set K c X and [or each e > 0 there is a continuous finite-rank operator A : X

X such that for all x e K, IIAx - xll < e.

1.24 T H E O R E M . Let X be a Banach space such that X* has the approximation

property. Then for any Banach space Y, the set o[ compact operators in r ( X , Y ) is the

closure of the set o[ operators having t~nite rank.

P R O O F . Suppose tha t A is compact; then its adjoint A* is also compact; see [57~

p. 485]. Take e > 0 and let

g = { A * ¢ : ¢ e Y * , II¢ll < 1}.

Then the closure of K is a compact set in X*, and so there exists a finite rank operator

F E ~ . (X*,X*) such tha t

(2) I [ ¢ - F ¢ I I < e for all ¢ E K .

Now FA* belongs to ~.(X**, Y**) and has finite-dimensional range. Hence its adjoint

A**F* belongs to /~(X**,Y**) and has finite-dimensional range. From Eq. (2), if x E

X, Ilxll = 1, and ¢ E K, then

I¢(:~) - (r¢)(x)I <- e .

I f ¢ e Y* and I1¢11 ~ 1, then a * ¢ e K and

ICA*¢)Cz)- (FA*¢)Cx)I < ,.

If J x : X --+ X** denotes the natural embedding, then

I (A**Jxz ) (¢) - ( A * * r * J x z ) ( ¢ ) I <_ ,.

By taking a supremum on ¢ we get

I I a * * J x z - A**F*Jxxl l < ,.

Since A is compact ,

by 11.12, and so

o r

A'*(x**) c J.(Y)

N J ~ I A * * J x x - J y I A * * F * J x x l l < e,

17

Taking a supremum on x yields

IIAx- jy1A**F*Jxxll ~ ~.

IIA- Jy1A**F*JxI f <_ e.

Thus J~. IA**F*Jx is the required operator of finite rank.

Conversely, if {F .} is a sequence of finite rank operators in L(X, Y) then each F .

is clearly compact. The set of compact operators is closed in the norm topology [57, p.

486], and so if iIA - r . l l --* 0 a s n - - , oo , a must also be compact. •

1.25 T H E O R E M . Let X and Y be Banach spaces. Then X* ®~ Y* is (isometrically

isomorphic to) the closure in the norm topology of the set of all operators from X to Y*

which are of finite rank.

P R O O F . Let ¢ ~ E X * a n d ¢ ~ E Y * , f o r l < i < n . The expression

F$

i = 1

defines an operator A from X to Y* by the equation

Ax : ~ ¢,(x)¢~ (x • X). i = 1

Equivalent expressions lead to identical operators A. It is clear that every operator of

finite rank from X to Y* can be obtained in this way from an expression in X* ® Y*. Now

define the operator B from X** to Y* by the equation

gt

Bp = ~ p(¢,)¢, (p • X**). i = 1

Furthermore,

and

A*q = ~ q(¢i)¢i i = 1

(q e Y**)

A**p = ~ P ( ¢ , ) J ¢ i (p E X**) i = l

where J is the canonical embedding of Y* in Y***. Hence J B = A** and B = J-1A**,

whence

18

1.26 T H E O R E M . Let X be a Banach space such that X* has the approximation

property. Then for any Banach space Y, X* ®x Y* is the space of alI compact operators

from X to Y*.

P R O O F . Use 1.24 and 1.25. 1

1.27 C O R O L L A R Y . Let X and Y be Banach spaces, it being assumed that X*

has the approximation property. In order that

X* ®x Y* = (X ®~ Y)*

it is necessary and sumcient that each element of ~(X, Y*) be compact.

P R O O F . Use 1.26 and 1.22. •

1.28 C O R O L L A R Y . Let p, q be real numbers satisfying 1 _< p < q < oo. Then

1 1 1 1 *q' ~A ep = (~q ~ 7 ~P')*' where ~7 + q p, + -p = 1.

P R O O F . A classical theorem [126, p. 76] asserts that if 1 < p < q < oe then every

member of £(£q, ¢'v) is compact . Thus in the previous corollary we take X = ~q, Y* = £p

and use the familiar identification

e;=t~,, l_<p<oo. •

1.29 D E F I N I T I O N . Let U, V, X, and Y be Banach spaces. / /

A e . C ( X , U ) and B e f . ( Y , V )

then A ® B is det~ned on X ® Y by

g t

t A G = ® , y , . /=1 i=1

Observe tha t (A ® B)(z) is independent of the representat ion of z. Indeed, if

then by the definition of %

~ Axi ~ Byl <_ i=l

i=1

tiA~,ll IIBy~II ~ IIAII IIBII ~ ll~t[ IiY~ll- i = 1 i=1

19

B y taking an infimum over the representations of z , we obtain

"I(A ® B)(z ) <<_ IIAII llBHq(z).

Thus if

i = l

then z - - 0 and ( A ® B ) z = O .

1.30 L E M M A . Let X and Y be Banach spaces, and let c~ be a uniform crossnorm

on X ® Y. I f A E ~ (X , X ) and B e ~(Y, Y), then A ® B has a unique continuous linear

extension, A ®a B, on X ®a Y-

P R O O F . By the uniform proper ty of c~,

a A x , ® U y ~ <i lAi i l iBi la x~®y~ . i = 1

This establishes that IIA ® BII < ]IAII IIBI]. (By considering appropr ia te dyads, one can

see tha t equality holds here.) By the s tandard theorem on extending bounded operators

defined on dense subsets [165] we obtain the extension, denoted by A ®~ B, and defined

on X ® ~ Y. Its norm is ]JAil HBII. •

Let X and Y be real Hilbert spaces, in which the inner products are denoted by (a, b).

We shall now consider the problem of constructing an inner product on X ® Y. A pseudo

inner product is defined by

(x®y, a®b) = (~,a) (y,b).

This is extended by linearity, so tha t if

u = ~ x, ® y, and v = ~ ai ® b , i = i j = l

then

(3) <~, v> = ~ ( x , , aj> (y,, bj>. i , j

The next three lemmas establish tha t Eq. (3) defines a genuine inner product on X ® ]i.

1.31 L E M M A . /T u ,,~ u t and v ~., v I then

(~', , '> = (~, ~>.

2O

P R O O F .

Suppose

We prove only (u, v) = (u ' , v). A s imi lar p roof shows t h a t (u ' , v) = (u ' , v ' ) .

rt k rrt

E ' ' E = x i ® Y i and v : a ~ ® b i . i : 1 i : l i : 1

Since u ~ u ' ,

for all ¢ E X*. Hence

so t h a t

k

~(~,)y, E ' ' = ¢(x,)y,

i = I i : I

n k

E ( a j , xi)y, = E < a i , x~)y~ for l _ < j _ < m i = l i = 1

tt rn k

j ~ l i ~ l 3"----1 i = l

and (~, ~) = (¢ , ~). m

1 .82 L E M M A . Equation (3) de~nes a positive semi-de~nite form on X ® Y.

P R O O F . We invoke the t heo rem [18, p. 422] t h a t a posi t ive l inear ope ra to r on a Hi lber t

space has a unique posi t ive square root . Consider the m a t r i x A whose e lements are

a~ i = ( x ~ , x j ) where x~ E X .

This m a t r i x is posi t ive semi-defini te since

~ (x~ , x3.)~ ~. = ~x~ , ~3xi _> 0 i , i = l i = 1 3"=1

for all vectors ( ~ 1 , . . . , ~,t). The m a t r i x A can then be wr i t t en in a unique way as A = B 2,

where B is also posi t ive semi-definite. Since A is symmet r i c , we have A = A T = (BT) 2

and conclude ( f rom the uniqueness of B) t ha t B -- B T. Now we wri te

r$

u = E xi ® yi and A = B B T i = l

so t h a t

(~, ~') = E ( ~ , , ~;) (y,, yJ) = X : ~,~<y,, y;) if ij

i f k

k " j

21

1.33 L E M M A . For an expression u = Exi ® yi, the fol lowing propert ies are equiv-

alent:

(i) (u ,v} = O for all v E X ® Y;

(ii) (u, u} = 0;

(iii) u = 0 as an operator f rom X to Y.

P R O O F . The impl ica t ion (i) ==~ (ii) is trivial. For the impl ica t ion (ii) = : ~ (iii), use

the Cauchy-Schwarz inequali ty to conclude tha t (u, v} = 0 for all v E X ® Y. If v = x ® b

then

0 = <u, ~> = ~(~,, ~> (y,, b> = <r~<~,, ~>y,, b).

Since b is arbi trary, ~(x i , x)yi = 0, showing tha t u is the zero operator . For the impl ica t ion

(iii) ==~ (i), suppose u = 0 as an operator . T h a t is, u -,~ 0. Taking u ' = 0 in 1.31, we

conclude tha t (u, v} = 0 for all v. •

Equa t ion (3) clearly defines a linear and symmet r i c bil inear form on X ® Y. By 1.32

and 1.33, this bilineax form is a genuine inner product . The no rm induced by this inner

p roduc t is

i.e., } i / 2

~ ( ~ x , ® y,) = ~ - ] (x , , ~j> (y,, yj> i ]

The comple t ion of X ® Y with norm ~ is denoted by X ®~ Y and is, of course, a Hilbert

space.

1 .34 L E M M A . The norm ~ is a reasonable crossnorm, and ~* = ft.

P R O O F . The crossnorm proper ty follows at once f rom the definition, since

~(~ ® y) = [<x,~> <y,y>]l/2 = Ilxll Ilyl[

The norm fl*, as defined in 1.3, is in this case

i = l

= sup

gt t

E,%~ Ei=~<~,, ~J> <y,, b>

i = i a j ® b 5

If

u = z i ® y i and v = a j ® b j i----I y=l

22

then

z'(,~) = sup !.u.,,,) = z(,,). • z(,,)

1 .35 L E M M A . For a n y operator L : X --* X we have

) ( ) E L = , ® y , _<llLtl i = l i = l

P R O O F . Define the n x n m a t r i x A = (a~y) where a~y = (y~, y]}. As in t he p r o o f of 1.32

A is pos i t ive semi-def in i te and has a f ac to r i za t i on A --- B B T. T h e n we can w r i t e

i iy iy

i y v " y

2 2 b i u x i 2 = E F.b,.L=, -- L b,.=, <- IILII2 E E i v i

u i y i y u

i y i

1 .36 L E M M A . The norm fl is a uniform crossnorm.

P R O O F . Using the p reced ing l e m m a and a s y m m e t r y a rgumen t , we have for any two

o p e r a t o r s L1 : X ---* X and L2 : Y -+ Y,

fl L l x , ® L2y~ <_ ]IL1Hfl x, ® nzy~ <_ I]L1H tln~Hfl z, ® y, . • - - i ---.--- 1 - - i = i

1 . 3 7 L E M M A . Let G and H be closed subspaces in Hilbert spaces X a n d Y re-

spectively. Then the Hilbert-space crossnorm fl on G ® H is the r e s t r i c t /on to G ® H o f

the Hilbert-space crossnorm on X ® Y.

P R O O F . Deno te the H i lbe r t - space c rossnorm on X ® Y by ft. Thus

~(~x,®y,)= E(~,,xj} (y,,yj) ~ , e x , y, eY . z3

The Hi lbe r t - space c rossnorm on G ® H is defined by

f l l ( E g , ® h , ) = ~-~.(g,,gy) (h, ,hy) g, e G , h, e H . ~3

23

It is therefore clear t ha t fll is the restr ict ion of fl to G ® H.

1 .38 L E M M A .

and de~ne

If

Let S and T be a-6nite measure spaces. Let

n

E x, ® y, E L2 (S) ® L2 (T) i = 1

n

/(s, t) = ~ ~,(s)~,(t). i = 1

Ex i®y i ~ 0 ® 0

then f(s , t) = 0 almost everywhere.

P R O O F . Let # and u be the measures on S and T respectively. Let a be the p roduc t

measure, # ® u. By the Fubini Theorem [148, p. 150] we have then (with z = Ex~ ® y~)

(4)

II'll ~= f ,~ ~ = f f ,:(~, ~ ) ~ = f f ~ ~,(~),, (~)~J(~),; ( ~ ) ~ i3'

~3

= ~ ( ~ , , ~;.) (y,, y;.) = ( ; , ; ) = Z:(z) . i3"

Now if

~ - ~ x ~ ® y ~ - ~ 0 × 0 , i----1

then by 1.33, (z ,z) = 0. Hence Ilfl] = 0, and f (s , t) = 0 a lmost everywhere.

1 .89 T H E O R E M . /1 S and T are a-/~nite measure spaces, then

L2 (S) ®~ L2 (T) = L2 (S x T).

P R O O F . Let z be a member of L2(S) ® L2(T), say

z = ~ xi ® yi. i = 1

We associate wi th z the funct ion f defined by

/ ( s , t) = ~ x,(s)y,(t). i = 1

24

II

By the preceding lemma, f does not depend on the representation chosen for z. Equation

(4) in the preceding proof shows that f 6 L2(S × T). The map z H f is linear, and by

Eq. (4) it is norm-preserving. Thus the completion of L2(S) ® L2(T) in the fl-norm is its

closure in L2(S × T). To complete the proof we show that L2(S) ® L2(T) has orthogonal

complement 0 in L2(S × T). To this end, let f E L2(S × T) and suppose that

( f , x ® y ) = O for all z e L 2 ( S ) and all yeL2(T) .

By the Fubini Theorem, we write this condition as

f Y(t) [ f x(s)ft(s)ds] dt=O.

Since y is arbi trary in L2 (T) we conclude that for almost all t,

f (s)f(s) as o Since x is arbi t rary in L2(S), we conclude that for almost all t, f~ is the 0-element in

L2(S). Hence

f I/(s,t)]2ds= f { f | 1.40 L E M M A . Let X and Y be Hilbert spaces, and let P be the orthogonal pro-

jection of X onto a subspace G. Then P ®a Iv is the orthogonal projection of X ®~ Y onto G ®~ Y.

P R O O F . It is clear that P ®a I maps X ®a Y into G ®~ Y and leaves invariant each

element of G ®a Y. For the orthogonality property we only need verify that

x ® y - ( P ® a I ) ( x ® y ) _ ] _ g e v ( x e X , y E Y , g e G , v E Y ) .

Computing the inner-product, we get

<(x-Px) e y , g e v > = ( x - P x , g) ( y , v ) = 0

because P is the orthogonal projection of X onto G. II

1.41 D E F I N I T I O N . h e A and B belong to f .(X,X) then their Boolean sums are

de~ned by A @ B = A + B - A B

B @ A = B + A - B A .

1.42 T H E O R E M . Let X and Y be two Hilbert spaces. Let P and Q be orthogonal

projections on X and Y respectively. Then

(P ®~ I) @ (I ®~ Q)

25

is the orthogonal projection of X ®t~ Y onto

.~(P) ®~ Y + X ®a .~(Q).

P R O O F . By 11.2, it remains only to prove that the given projection has the orthogo-

nality property. It is to be proved that for arbitrary z • X ®fl Y and arbi t rary

w • ]~(P) ®~ Y + X ® ~ ;~(Q)

we have

The case q = oo is given by

z - [(P ® I) @ ( I ® Q)lz ± w.

It suffices to prove this for w E ~ ( P ) ®~ Y and then use a symmetry argument. Because

of linearity and continuity it suffices to verify the orthogonality relationship for dyads:

z = x ® y and w = g ® v , where x E X , y E Y , g E . ~ ( P ) and v E Y .

Calculating the required inner product, we have

( x ® y - P x ® y - x ® Q y + P x ® Q y , g®v>

= <x, g> <y, v) - (Px, g> (y, v) - <x, g> <Qy, v> + (Px, g> (Qy, v)

= <x-Px, g> ( y , v > - ( x - P x , g) (Qy, v> =0

because x -- P x ± g. I

In previous theorems (viz. 1.14, 1.16, and 1.39) it has been shown that

(i) C(S) ®x C(T) = C(S x T)

(ii) L1 (S) ®.~ nl (T) = L1 (S x T)

(iii) L2 (S) ®f~ L2 (T) = L2 (S x T).

In the remainder of this chapter, the analogous result for Lp(S x T), 1 < p < oo,

will be developed.

1.43 DEFINITION. Let Y be a Banach space. For YI, ... , Y, E Y and 1 <_ q < oo,

we define

} [¢(Y,)] q : ¢ • Y*, ]1¢[[ --- 1 . i-----1

# ~ ( Y l , . . . , Y , ) - - - - s u p { max ] ¢ ( y , ) ] : ¢ e Y * , ]]¢]]-- 1}. l < i < n

26

1 . 4 4 L E M M A . Let l < p ,q <_ oo and p - l + q-1 = 1. Then

\ 1/v < 1~. ) i = 1

P R O O I ~ . We fix the n-tuple (Yl , - . . , Yn), with Yi E Y. Let

= ( ~ 1 , . . . , ~ , ) e R " , w i t h II~llp = (r~l~ , lP) 1 / ' = 1.

For each ¢ e Y* with IICH = 1, we set ¢ = (¢ (Yl ) , - - - , ¢ (Y, ) ) . Thus ¢ E IR". By using

the duality between ~ and ~ , we have (from 1.43)

~ , d w , . . . , v,,) = s~p I1~1t~ = supsup(A, ¢)

= sup sup ~ Ai ¢(yi) = sup sup ¢ A~ y~

=supx A~y~ . II

1

p - l + q - 1 = 1:

~pC~) in f f1~,11 p' 1/~ = ~ q ( v l , . . . , v . )

1.45 D E F I N I T I O N . Let z E X ® Y, where X and Y are Banach spaces. For

<_ p <_ oo, the p-nuclear norm o f z is defined by the fol lowing equation, in which

: z = ~ z ~ ® y i } . i = 1

The in f imum is taken wi th respect to all represen rations o f z, and (E I Ai [P)1/p is unders tood

to mean max fail when p --- oo.

1.46 L E M M A . The p-nuclear norm is a reasonable crossnorm.

r~ P R O O F . Let z E X ® Y, and let one of its representations be z -- ~ = 1 z~ ® yi. Let

A be the operator from Y* to X corresponding to z. Then for all ¢ of norm 1 in Y*, we

have, by the HSlder inequality,

]IA¢[[ = ~-~¢(y,)x, <_ I¢(Y,)III~,II i----1 i----1

- ,=1 ,=1 <_ , q ( v l , . . . , v , ) ,=1 ~ llz'll'~ 1/,,

In this inequality, take an infimum over all representations of z, followed by a supremum

in ¢. The result is

IIAII _< ap(z ) .

27

This shows that if z # 0, then A # 0 and ap(z) > O.

The triangle inequality for ap is proved as follows. We want to show that for all e > 0,

If e is given, we select a representation for z, say

$%

z = E Xi ® Yi, i----1

such that

I1~,11" , , (y~, . . . ,y ,~) < ~,,,(~)+ ~. i : l

Since 1/p + 1/q = 1, and since the factors on the left in the preceding inequality are

homogeneous, we can assume further that

and that

,q (y l , . . . , y.) _< [~p(z) + ~]'/,.

Then for all functionals ¢ of norm 1 in Y* we have

Obviously then,

and

l ip

] [~-~ i¢(y,)l ~ ~/~ L,=I < [~,:.(z) + ~]l/q.

n

I1~,11 p ~ o~,oCz) +,~

r$

I¢(y, ) l q ~ ,~ (~) + ~. i = l

W

Now take a representation for w, say

with the analogous properties. Thus

i = n + l

and

i = n + l

I1~,11 ~' ~ ,~p(~) + ~

28

Then

and

Consequently

m

I¢(y,)l q _ ~p(~) + ~. i=n+l

II~,It p < ~ (~ ) + ~ ( ~ ) ÷ 2~

I¢(y,)l ~ < ~,(~) + ~, (~) ÷ 2~. i=l

[ .LL ]1,. x, ll p y, _< ~p(z) + ~p(~) + 2~.

i----1 '= * 1

Upon taking a supremum in ¢, we obtain the desired inequality.

For z = x ® y, we have, immediately from 1.45,

(5) ap(x @ Y) -< (llxll~)~/%q(Y) = llxll IlylI.

Now let ¢ E X*, ¢ E Y*, and z = ~ = 1 x~ ® Yl. Then

(¢ ~ ¢)(z) = ~ ¢(~,)¢(y,) <- II¢II II~,ll l¢(y,)l i=1 i=1

i ~ l =

~-~ t]¢ll II¢ll ( ~ NXiI'p) I/p#q(Yl'''''Yn'"

By taking an infimum over all the representations of z, we obtain

(¢ ~ ¢)(~) -< I1¢11 II¢lI-Az).

By taking a supremum over all z for which ap(z) _< 1 we obtain, by 1.3,

(6) ~ ; (¢ ® ¢) < il¢II II¢II.

From Eqs. (5) and (6) and 1.4, ap is a reasonable crossnorm. •

1.47 L E M M A . Let Y be a Banach space, S a ~nite measure space, and 1 < p < oo.

The natural embedding of Lp(S) ®.p Y into Lp(S, Y ) is of norm 1.

P R O O F . Let n = ~ ~, ~ y, e L~(s) ® v.

i=l

29

The natura l embedding associates with z the function f defined by f ( s ) = Ei"=l xi(s)yi .

It is to be shown that Ilfl[ -< ap(z) . If 1 < p < co, then

liflb = (

_-{

Ili(~)ll p ~ = ~ , ( 4 y , S S i = 1

Vds y , sup ~,(~)¢(y,) (¢ e * II¢II = 1) S ¢ i = 1

sup ~ I~,(~)1" [¢(yj)l q ds S ¢ i----1

11~,11 ~ . ~ ( y l , . . . , y . ) . i = l

F o r p = o o , w e write

II/11 ~ = ess s u p II f(s)II = ess s u p ~ xi (s)yi 8 8 i = 1

= e s s s u p s u p ¢ ~ x , ( ~ ) V , e s s s u p s u p x i ( 8 ) ¢ ( y i ) .q ~b i = 1 8 ¢ i = 1

< e s s supsup max Ixi(s)l I¢(y,)/ s ¢ i=l

= ( e s s s u p m a x ]xi(s)[} ( s u p E ]¢(Y¢)] I, 8 * ) l, ¢ i = l

= max I1~,11 m C w , . . . , v , ) . i

Thus for 1 g p g co we have

t l f lb ~ I1:~,11" ,%( .v l , . . . ,v , ) .

By taking an infimum over all representations of z, we have by 1.45,

II.rlb ~ o,,,(~).

The embedding is the (unique) continuous extension of the map z ~-* f defined above. In

order to see tha t the embedding is of norm 1, let z be a dyad, z = x ® y, in the preceding

computa t ion . One sees at once tha t

IlflJ = 11~11 llvlI = % ( z ) . •

30

1 .48 L E M M A . I f x i , . . . , z , E Lv(S) , l _ < p < o o a n d x i x 5 =O when i # j , then

P R O O F . Let

E, = {8 e s : 54(8) # 0} .

Then by 1.44,

} ~ a ( ~ , . . . , ~ , ) = s n p ~j5jll : I1~11~=1 A 3 1

{If "}'" = ~up ~ j x i ( 8 ) d~ ,X ]=1 .}1,.

= sup ,kdxj(8) ds ~t

= s u p In, l" 115411" A i=I }1,.

= snp oi ft~II ~ ~lO, l=i _

= {m~x. II~,IIU 1/~ = maxi 115411. I

1 .49 L E M M A . Let Y be a Banach space, S a finite m e a s u r e space, and 1 <_ p < oo.

The naturaI embedding of Lp(S, Y ) into Y ® ~ Lp(S) is of norm 1.

P R O O F . Let f be a s imple func t ion in Lp(S, Y) . Then f can be w r i t t e n in the form

n

s(~) = ~ 5,(8)y,, i = l

where the func t ions xl are mul t ip les of cha rac te r i s t i c funct ions . We can assume t h a t

x ix j = 0 if i # j and t h a t Ilxill = 1 for 1 < i < n. The image of f u n d e r the n a t u r a l

e m b e d d i n g is the e lement w = ~i~=1 Yl ® xi. I t is to be shown t h a t a v ( w ) < IIf[]- By 1.48,

g q ( x l , . . . , x , ) = 1.

Hence by the def in i t ion of av(w) in 1.45,

°.,w,<

31

On the other hand, if E~ = {~: z , ( s ) ¢ 0}, then

IlfH = { / ll/(~)llp d~} lip

}'" = IIx,(~)y, II p ds

} 1/p = ~Ny, llp/lz,(s)lpds i = 1

= ly, llpj i = l

Thus we conclude that ap(W) < I]f[[. The simple functions form a dense set in Lp(S,Y)

by [57, p. 125]. Hence the embedding (as defined thus far) has a unique linear extension

of norm at m o s t 1. If f in the previous argument is a constant funct ion, say f ( s ) = y,

then w = y ® 1 and

~ p ( ~ ) = Ityll lilt{ = N/II.

Hence the embedding is of norm 1. •

1.50 T H E O R E M . Let S be a ~nite measure space, and 1 <_ p < oo. Let Y be a

Banach space such that, under the natural map,

Lp(S) ®.p Y = r @.p Lp(S).

Then Lp(S) ®,p Y = Lp(S, Y).

P R O O F . Let F be the natural embedding of Lp(S,Y) into Y Nap Lp(S). By 1.49,

IlrI[ -- 1. Let ¢ be the natural map of Y Gap np(s) onto Lp(S) Gap Y. By hypothesis, ¢

is an isometry. Let h be the natural embedding of Lp(S) Gap Y into Lv(S , Y). By 1.47,

IIAI{ = 1. We will prove that

ACF = I.

It suffices to prove that A,I, r l = f for any simple function f in Lp(S, Y). Let f (s) =

~ = 1 c~(s)y~, where y~ e Y and the c~ are characteristic functions of measurable sets in

S. Then

i = l i----1

= F_, c,(~)y, : f(s) i = 1

32

The map CF is the required isometry, since

I l f l l -- I IACL"f l l _< I lOrf l l ~ Ilfll. m

R e m a r k . Under the conditions of the theorem, the following diagram is commutative:

Lp(S,Y) r y @.p Lv(S )

'I 1 ° Lp(s,Y) A L~(S) ®~, Y

1.51 L E M M A . Under the natural map, we have

Lp(S) ® ~ Lp(T) = Lp(T) ®~p Lp(S) (1 < p < oo).

P R O O F . Let z = ~ xi ® y~ E Lp(S) ® Lp(T). The natural map associates with z the n

element u = ~ i = 1 Yi @ xi in Lp(T ) ® Lp(S). We want to show that

~ , , ( z ) = ~,,(,~).

Because of symmetry, it suffices to prove that

~p(~) < ~p(z).

With z, we associate f E Lp(S, Lp(T)) by means of the natural map of 1.47. Thus, by

1.47, ~ t

f(~) = ~-: ~,(~)y/ a .d II/11 -< ~p(z). i = 1

With f, we associate w E Lp(T) ®~p Lp(S) by means of the natural map of 1.49. Thus,

by 1.49

= ~ y, ® ~, and ~,~(~) <_ Ilftl. i = 1

Since w = u, we have established that sp(u ) <_ ap(z). •

1.52 C O R O L L A R Y . For 1 < p < oo,

nv(s ) ® ~ np(T) = Lp(S × T).

P R O O F . By 1.51, the space Lp(T) has the property required of Y in 1.50. Hence,

np(s) ® ~ Lp(T) = Lv(S , np(v)).

33

It only remains to prove that

Lv(S, Lp(T)) ---- Lp(S x T).

With any f e Lp(S, Lp(T)) we associate z e Lv(S × T) by the definition

zC~, t ) = [ f ( ~ ) ] ( t )

It is routine to verify that every element of Lp(S x T) arises in this way from a suitable

f • Lp(S, Lp(T)) , and that

I1=11 = j I= ( , , t ) l '> - - j I IS(~)II ' = IlSlt.

(The Fubini theorem is used in this last calculation). •

1.53 T H E O R E M . Let S and T be a-/~nite measure spaces. Then L ~ ( S ) ®), L~, (T)

is a subspace of Loo(S × T). This subspace is usually proper.

P R O O F . Let X = LI (S) and Y = LI(T) . Then X* = L ~ ( S ) and Y* = L ~ ( T ) . Now

by 1.22 and 1.25, we have

(7) X* ®x Y* = FR(X , Y*) • £(X, Y*) = (X ®., Y)*

where F R denotes the closure of the set of finite-rank operators. Using 1.16, we have

Loo(S) ®~ Zoo(T) c [LI(S) ®7 LI(T)]* = LI (S x T)* = Loo(S x T).

The inclusion in (7) is usually a proper one. •

34

C H A P T E R 2

P R O X I M I N A L I T Y

Let K be a subset of a Banach space X. We say that K is p r o x i m i n a l in X if to each

x in X there corresponds at least one point/co • g such that I I x - k0l[ = infkeg [Ix -- kll.

The point k0 is then called a b e s t a p p r a x i m a t i o n t o x from K. Some questions about

proximinality in tensor product spaces now suggest themselves. Let X and Y be Banach

spaces and let H be a subspace in Y. Under what conditions can it be asserted that X ~ H

is proximinal in X®o Y? In particular, is the proximinality of H in Y a sufficient condition?

This question is addressed in the first half of the chapter. (The answer depends on the

crossnorm ~.) In the second half, the central question is whether (G ~ Y) + (X ~ H)

is proximinal in X ®a Y, it being assumed that G and H are subspaces of X and Y

respectively.

If K is proximinal in X, then any map which associates with each element of X one of

its best approximations in K is called a p r o x i m i t y m a p . If K is such that each element

of X has exactly one best approximation in K, then K is said to have the C h e b y s h e v

property.

If X is the space C(S) and if a is taken to be the injective tensor norm A, then an

easy theorem is available.

2.1 T H E O R E M . Let S be a compact Hausdorff space, and let H be a subspace of

the Banach space Y . / f there is a continuous proximity map of Y onto H, then C(S) ~ H

is proximinal in C(S) ®x Y, and in fact it has a continuous proximity map.

P R O O F . Let A : Y--~H be a continuous proximity map. Then the mapping A' :

C(S, Y) ---* C(S, H) defined by A' f = A o f is a proximity map. The continuity of A'

follows from 11.8. If g • C(S, H) then

for M1 8 E S. Hence

I l l ( s ) - A(f(s))ii <_ fir(a) - g(s)li

I l l - A o f l i ~ I l l - gll-

Now use 1.14, which states that C(S ,Y) = C(S) ®), Y, and 1.17, which states that

C(S) ®x H = C(S) ~ H. |

2.2 COROLLARY. Let H be a subspace of a Banach space Y. If either

(i) H is ~nite-diraensional and Chebyshev; or

Oi) Y is uniformly convex,

then C(S, H) is proximinal in C(S, Y).

P R O O F . In bo th of these cases, a theorem from [104, p. 164] guarantees the continuity

of the proximity map from Y onto H. The preceding theorem then applies. •

An n-dimensional subspace H in C(T) is called a H a a r subspace if no function in

H \ 0 has n zeros.

2.8 C O R O L L A R Y . f f H is a 6nite.dimensional Haar subspace in C(T), then

C(S) ®x H is proximinal in C(S x T).

P R O O F . Every Haax subspace has the Chebyshev proper ty [104, p. 114]. Hence by

2.2, C(S) ®:~ g is proximinal in C(S) ®x C(T). The lat ter is C(S x T) by 1.14. I

2.4 T H E O R E M . Let H be a closed subspace of the Banach space Y. Let S be a

compact Hausdorff space. For each f E C(S, Y) we have

dist (f, C(S, H)) = dist (f , eoo (S, H) ) = sup dist ( f ( s ) , H) . 8

P R O O F . Since C(S,H) c e~(S,H) it is clear tha t

dist ( f ,C(S,H)) ~ dist ( f ,g~(S,H)) .

If g E £~ (S, H) then for each s E S,

It follows tha t

and tha t

II.fCs) - g(s)li dist (y ( s ) , H) .

ll.f - gll sup dist ( f ( s ) , H ) S

dist (f , £~ (S , H)) _~ sup dist ( f ( s ) , H) . 8

For the reverse inequality, let A > sup d is t ( f (s ) , H.) For each s E S define S

~(s) = {h e H : tlf(s) - hll _< ~}.

Then (I)(s) is a nonvoid, closed, convex subset of H. We shall prove tha t ¢ is lower

semicontinuous. Let 0 be an open set in H and put

36

It is to be shown tha t 0* is open. Let a 6 0 " . Then ¢ ( a ) N O is nonvoid. Hence there

exists an h 6 ¢ such tha t [If(a) - hll < A. By the definition of A, there exists an h ' 6 H

such tha t [[f(a) - h'[[ < A. By convexity, there exists h" a O such tha t f ir(a) - h"[[ < A.

Let ~ / b e a ne ighborhood of a such tha t

t t r ( 8 ) - f ( o - ) l l < ~ - 11 t (o9 - h " l l

for all s E A/. For any 8 @ J~ we then have

Jl f ( .s) - h " l [ _< I I f ( .~) - J ' (o) lJ + l l f ( o ) - h " l l < ,x.

Hence h" 6 ¢ ( s ) N 0 , s e 0 " , ) / c 0 " , and 0 is open.

By the Michael Selection Theorem (11.14) there exists g 6 C(S, H) such tha t g(s) 6

¢(8) for all s. Hence [If(s) - g(s)[[ < A and I[f - g[[ <- A. It follows tha t

dist (f, C(S, H)) < A

and

dist ( f , C ( S , H ) ) <_ supd i s t ( f ( s ) , g } . • s

2 .5 C O R O L L A R Y . Let H be a closed subspace of Y. In order that an element h of

C(S, H) be a best approximation to an element f of C(S, Y) it is necessary and sumcient

that for some point so 6 S, so is a maximum point of [[f(s) - h(s)I I and h(so) is a best

approximation to/(so) from H.

2.6 T H E O R E M . Let S be an infinite compact metric space, and let T be a non-

atomic measure space. / f H is a finite-dimensional subspace of L1 (T), then CCS, H) is not

proximinal in C ( S , Lz ( T) ) .

For a p roof of this, as well as related results, see [61].

2 .7 T H E O R E M . Let S and T be closed and bounded intervals of the real line. Then

there exists a one-dimensionM subspace H in C(T) such that C(S) ® H is not proximinal

in C(S x T).

P R O O F . The cons t ruc t ion and proof are s impler ff we use S = [ - 1 / 2 , 1/2] and T =

[--1, 1]. Define

h(~) = t

r ( ~ , t ) = 1 - Is - t i

F ( ~ , t) = .rCs, ~:)/(1 - I.-"1)

~eT

(8, t) e S × T

(s,t) e S x T.

37

One can show tha t the best a p p r o x i m a t i o n to fe is h if s > 0 and - h if s < 0. F rom

this, dist {fs, H} = 1 - Ish where H denotes the subspace genera ted by h. It then follows

t h a t dist ( F ~ , H ) = 1 for all s, and dist ( F , C ( S ) ® H ) = 1. But if x e ~oo(S) and

H E - x ® hi[ < 1, then x(s) = ( 1 - s) -1 for s > 0 and x(s) = - ( 1 + s) -1 for s < 0. T h u s

x mus t be d iscont inuous a t s = 0. The tedious detai ls are omi t t ed . •

2 .8 T H E O R E M . Let (S, A,/z) be a finite, complete m e a s u r e space. Let G be

a t~nite-dimensional subspace in a Banach space X. Then Loo(S, G) is proximinal in

Loo (S, X).

P R O O F . Let f 6 Loo(S,X). For each s C S define

¢( s ) = {g e G : II/(s) - gll = dist ( f (8) , G)} .

For each s, ~(s) is a closed, bounded , and n o n e m p t y subset of G. We shall show tha t

is weakly measu rab le in the sense t h a t for each c o m p a c t subset K of G the set

K* = {s e s : ¢ ( s ) n K ¢ n}

is measu rab le in S. The set K* can also be descr ibed as

K* = {s E S : inf ]If(s) - gll ---- inf Ill(s) - g[I}. g6K g6G

Since (S, ~, #) is complete, f is measurable in the classical sense, by 10.2. Since subtraction

in X and the n o r m in X are cont inuous, the m a p p i n g s ~-* II f ( s ) - gll is measurab le for each

g. Hence the m a p p i n g s ~ infae A Ill(s) -ell is measu rab le for any set A. It follows t h a t K*

is measurab le . By 11.17, there is a measurab le funct ion h : S --~ G such t h a t h(s) E ~ ( s )

for each s 6 S. Since G is f ini te-dimensional , it is separable . Hence by 10.3, h is s t rongly

measurab le . Since IIh(s)l I <_ 211f(s)ll, it follows t h a t Ilhll _< 211fl 1. Thus h e Loo(S, G). For

a n y k e Loo(S,H)we have l l f(s)-h(s)l t < l I / ( s ) - k ( s ) l l for all s. Hence I I / - h l l _< l lY-kl l -

Th is proves t h a t h is a bes t a p p r o x i m a t i o n to f in Loo (S, G). •

T h e previous t heo rem and the next do not fit into the tensor p r o d u c t f r amework , since

the space Loo (S, X ) is general ly not i sometr ic to the space Loo (S) ®x X. For example ,

•oo ®x el is a p rope r subspace of too(IN, el). In order to see this, observe t h a t by 1.26

£oo ®x gl is the subspace of c o m p a c t ope ra to r s in /~(£1, £1)- On the o the r hand , £oo (1N, el) conta ins /~ (£1, £1)-

2 .9 T H E O R E M . Let S be an arbitrary set, Y a Banach space, and H a proximinal

subspace in Y. Then £oo (S, H) is proximinal in too (S, Y).

P R O O F . If f E ~oo(S,Y) then one of its best app rox ima t ions is the funct ion g such

t h a t I l l(s) - g(8)[[ = dist ( f (s ) ,H) for each s e S. •

38

2 .10 L E M M A . Let (S,A,#) be a measu re apace, Y a Banach apace, and H a

subspace of Y. Then for each f E L I (S , Y )

(list ( L LI (S, H) ) = Is (list ( f ( s ) , H)&.

P R O O F . For any g E L1 (S, H ) we have, by the def in i t ion in C h a p t e r 10,

IlS-~I1 = f~ llS(~)- 9(~)llas >_ f~ dist (f(8),H)ds.

By t a k i n g an in f imum on g we o b t a i n

(list ($, LI(S, U)) > fs (list ( f (~ ) , H)ds.

For the reverse inequal i ty , let e > 0, and let f ' be a s imple func t ion in LI (S ,Y ) such

t h a t [If - f ' l l < e. Wr i t e f ' = ~.~in=l xiyi where the xi are the cha rac te r i s t i c func t ions of

sets Ai in d and Yi 6 Y. We m a y assume t h a t ~ n = i=1 xi 1 and t h a t /z(Ai) > 0. Since

f ' E L I (S ,Y ) , we have [[yil[/z(A,) < co for 1 < i < n. I f / z (A i ) < co, select hi E H so t h a t

I]Yi - hill < (list (yi,H) + n#(A,----~"

r$ I f / z ( A i ) = oo, pu t hi = 0. Let g = ~i=1 xihi. Then g E LI(S ,H) and

dis t (f ' , LI(S, H))

I l l - gll

f fir(s) - g(s)ll&

- - , + ~-~ f~ lly, - h, ll& i = 1 i

< e + ~ ilY~ - h, lt#(A,) i = 1

i = 1

= 2, + f d is t ( i f ( s ) , H)ds

÷ ft , t (:(,), .) ÷ llS(s) - &

dis t (f, LI (S ,H)) < e-t-

< e +

~ E q -

39

2.11 C O R O L L A R Y . Let H be a closed subspace of the Banach space Y. In order

that an element g of LI(S, H) be a best approximation to an element f of LI(S, Y) it is

necessary and sufficient that , for almost all s E S, g(s) be a best approximation in H to

2.12 L E M M A . Let S be an arbitrary measure space, and let H be a proximinal

subspace of a Banach space Y. Then each simple function f in L1 (S, Y ) has a best ap-

proximation w in L1 ( S, H) such that for all measurable sets A,

/A HW(S)[[ds ~-- 2 /A ['f(s)[[ ds"

P R O O F . The simple function f can be writ ten as f -- ~ i = 1 xl ® Yi where each xi is

the characteristic function of a measurable set Ai in S, yi E Y, and ~i~=1 xi -- 1. Let hi

be a best approximat ion to Yi in H, and set w = ~i=1 xi ® hi. For any v E L I ( S , H ) we

have

i = 1

i = 1 i = 1

= II zi(s)w - ~ - ~ z i ( s ) h i l [ d 8 = I l l - 11- i = 1 i = 1

Observe also tha t for each s, xi(s)w is a best approximat ion to xi (s ) f . Hence II~i(s)~oll _<

211zi(s)Yll. When this inequality is integrated over an arbi t rary set A, the inequality in

the Lemma is obtained. II

2 .13 : r I - I ]~OREM. Let H be a reflexive subspace of the Banach space Y. Let S be

a finite measure space. Then L1 (S, H) is proximinal in LI (S, Y) .

P R O O F . Let f be an element of LI (S , Y). Then there exists a sequence of simple

functions f,~ in LI (S , Y) converging to f . By 2.12, each f,~ has a best approximat ion w,~

in L1 (S, H) , and we may assume tha t

(1) fE II o( )ll for all measurable E c S. The sequence {f, h , h , . . . } is norm-compact and hence weakly

compact . It is therefore uniformly integrable by Theorem 4, page 104 of [55]. This means

tha t to each e > 0 there corresponds a 6 > 0 such tha t

f II.r,~Ks)ll& < ¢ whenever # ( E ) < 5 and n E ]N. (2) JE

40

By inequalities (1) and (2), the sequence {w,} is uniformly integrable. It is also bounded.

Hence by the Dunford compactness theorem [55, p. 101] the sequence {wr,} has compact

closure in the weak topology of L1 (S, H). By taking subsequences and using the Eber-

lein theorem, we may assume that w, converges weakly in L~(S, H) to an element w of

LI(S , H). It therefore converges weakly in L~(S, Y) to w [57, p. 436, Problem 6]. By the

weak Iower-semicontinuity of the norm,

Ill - ~oll l iminf Ill . - w.][ = l iminfdis t ( f . , L I ( S , H ) ) = dist (f , L I ( S , H ) ) . •

2.14 L E M M A . If P is a projection defined on a Banach space X, Shen She range

and null space of P* are proximinal subspaces in X*.

PROOF. It is elementary to prove that

P*(X*) = (ker P) ±.

Since the annihilator of a subspace of X is weak*-closed in X*, we conclude that the range

of P* is weak*-closed and proximinal. Now use the fact that ke rP* = ( I - P*)(X*) to

see that ker P* is weak*-closed. •

2 .15 L E M M A .

subspace

Let X and Y be Banach spaces. If P is a projection on X then the

M = { A P : A E 2~(X, Y*)}

is complemented, weak *-closed, and proximinal in £ ( X, Y * ).

P R O O F . Theorem 1.22 asserts that 2.(X,Y*) = ( X ® ~ Y)*. The space ~ ( X , Y * ) has

therefore a weak*-topology induced by its duality with X ®~ Y. In this topology, a net Aa

converges to 0 if and only if (A , x ) ( y ) -* 0 for all x E X and y C ]i.

For A E ~(X, Y*) define p(A) ---- AP. It is easily seen that p is a projection of 2.(X, Y*)

onto M. Now p is weak*-continuous because if As --~ 0 then p(Aa) --+ 0 since

(p(A~)x)(y) = (A , Px)(y) -* 0

for all (x, y). It follows that I - p is weak*-continuous and that its null space, which is M,

is weak*-closed. The proximinality of M follows from the fact that every weak*-closed set

in a conjugate space is proximinal [104, p. 123]. •

41

2.16 L E M M A .

the subspace

In the same way one proves a dual result:

Let X and Y be Banach spaces. Let Q be a projection on Y. Then

N = { Q ' A : A E £ . (X ,Y*)}

is complemented, weak*-cIosed, and proximinal in ~. (X, Y*).

2.17 T H E O R E M . Let P and Q be projections on Banach spaces X and Y respec-

tively. Then the subspace

W = { A P + Q ' B : A , B E / ' (X, Y*)}

is complemented, weak*-closed, and proximinal in ~(X, Y*).

P R O O F . The subspace W is M + N, where M and N are defined in the two preceding

lemmas. The projections p and q, defined by p(A) ---- A P and q(A) = Q ' A , have ranges

M and N. They commute with each other, since

p(q(A)) = (Q*A)P = Q*(AP) = q(p(A)).

Thus the Boolean sum p@q = p + q - p q is a projection of ~(X, Y*) onto W, by 11.1. Hence

W is complemented and norm-closed. By 11.3, W is weak*-closed and proximinal. •

2 .18 T H E O R E M . Let G and H be 6nite-dimenslonal subspaces in conjugate Ba-

nach spaces X* and Y* respectively. Then X * ® H + G ® Y * is complemented, weak*-closed,

and proximinal in ~ (X , Y*). It is therefore complemented and proximinal in X* ®~ Y*.

P R O O F . We prove first that for an appropriate projection Q on Y, we have

X* ® H = {Q*A : A E r_(X, Y*)}.

Since H is finite-dimensional, there is a projection Q on Y such tha t Q* (Y*) = H. (Select a

m h basis h i , . . . , hm for H and select y~ in Y so tha t y~(hj) = •j. Then let Q --- ~ = 1 ~ ®Y~)-

If N denotes the subspace of all Q*A for A E ~ ( X , Y * ) , then X* ® H c N. Indeed,

Q* o (~o ® h) = ~ ® h for all ~ ® h E X* ® H. For the reverse inclusion, note that each

element of N belongs to ~(X, g ) . Since g is finite-dimensional, ~ (X , H) = X* ® H.

A similar argument applies to show that

G ® Y* = { A P : A E £ ( X , Y * ) } .

Then an application of 2.17 completes the proof. •

42

2.19 COROLLARY. Let S and T be a-~nite measure spaces. If G and H are

finite-dimensional subspaces in Loo ( S) and Loo ( T) respectively, then the subspace

Loo (S) ® H + G ® Loo (T)

is weak*-closed, compIemented, and proximinal in Loo(S × T).

P R O O F . Let X = LI(S), Y = LI(T), and W ---- X* ® H + G ® Y * . Then X* --

Loo (S) and Y* -- Loo (T). By the preceding result, W is weak*-closed, complemented, and

proximinal in f . (X,Y*) . By 1.22 and 1.16,

f . (X,Y*) -~ ( X ® ~ Y)* = (LI(S) ®~ LI(T))*

= L I ( S x T ) * = L o o ( S × T ) . •

2.20 C O R O L L A R Y . Let S and T be arbitrary sets, and let G and H be t~nite-

dimensional subspaces in ~oo (S) and ~oo (T) respectively. Then

goo(S) ® H + G ® ~oo(T)

is complemented, weak*-closed, and proximinal in ~oo (S × T).

P R O O F . This is a special case of the preceding result, taking counting measure on S

and T. The a-finiteness assumption is not needed, as gl (S)* --- ~oo (S) for arbi t rary sets

[ 1 0 3 , p . 1 2 9 ] . •

2.21 T H E O R E M . Let U and V be proximlnal subspaces in a Banach space X.

Assume that U W V is closed, and that V has a proximity map A such that for each c E X,

the m a p u ~-* A(c - u) is weakly compact from U to V. Then U + V is proximinal.

P R O O F . Let c b e any element o f X , andse lec t z , E U + V so that

tfc - z,[[ ~ dist (c,U + V).

The sequence {zn} is bounded. Since U+V is closed, 11.3 implies tha t z,~ can be expressed

as u,~ + v,~, with u,~ E U, v , 6 V, and {u ,} bounded. Put v~ = A(c - u ,) . Since {u,~} is

bounded, our hypotheses imply tha t {v~} lies in a weakly compact subset K of V.

By the Hahn-Banach theorem,

dist (x,U) = s u p { ~ ( x ) : ~ @ V ±, H~[[ = 1}.

Hence the function x F-~ dist (x, U) is weakly lower semicontinuous, being the supremum

of a family of weakly continuous functions [148, p. 39]. Let v be a point in K where

dist (c - v, U) is a minimum. Let u be a best approximat ion of c - v in U. Then

I ] c - v - u[[-- dist (c - v,U) < dist (c - v: ,U)

t _< lie - . . - .ll -< lic - v . - .II.

43

By tak ing the l imit , we get IIc - . - ~tl -< dist (c, U -t- V). •

2 .22 T H E O R E M . Let G be a finite-dimensional subspace of C(S) such that

O ® C(T) is proximinal in C(S x T). Let H be a finite-dimensional subspace of C(T)

having a nipschi~z proximity map. Then C(S) ® H + G ® C(T) is proximinaI in C(S x T).

P R O O F . By 2.1, the subspace V = C(S) ® H is proximinal . Set U = G ® C(T) . By

11.2, U + V is c o m p l e m e n t e d and ('losed. Let A be a Lipschitz p rox imi ty m a p of C(T)

onto H, and define (A 'z ) ( s , t) = (Azs)(t). Then A' is a p rox imi ty m a p of C(S × T) onto

V. Define F : U --+ V by pu t t i ng F(u) = A'(z - u), where z is now fixed. By the following

l e m m a , r is (weakly) compac t . By 2.21, U + V is proximinal . •

2 .23 L E M M A . Let G and H be finite-dimensional subspaces of C(S) and C(T)

respectively. Let U = G ® C(T) and V -- C(S) ® H. Let J : C(T) ~ H be a Lipschitz

map. Then for each z E C(S × T) ~he map F : U ---* V defined by ~he following equation

is compact:

( r u ) ( s , t ) = (J(zs - us))(t).

P R O O F . Let B = {u C U : Ilull < k}. We will show tha t the closure of r ( B ) is compact

in V. By the Ascoli Theo rem, it suffices to prove t ha t F(B) is bounded and equicont inuous.

Since J is a Lipschitz map , there is a cons tan t A such t h a t

IIJ: - Jyll <- 11=- vii-

Hence, for any x • C(T)

nJxll <_. I I J x - JOll + IIJOl[

<- ,11 11 + IIJoll. Thus F ( B ) is bounded . The r ema inde r of the proof concerns the equicont inui ty of F (B) .

Select a basis {g l , . - - , g ,~} for G and funct ionals Vpl, . . . , !o ,~ e C(S)* so t h a t Ilg, II =

IlCsll = I and ¢,(gA = 6,; (11.11). If u E B, then u = E i = l gi ® yi and

Let (a, r) be a poin t of S x T at which the equicont inuiy of F (B) is to be proved.

Let e > 0. By the equicont inui ty of the uni t sphere in G there is a ne ighborhood $/1

of a such t h a t lg(s) - g(a)l < ellgll for all g e G and all s e $/~. Similarly, there is a

ne ighborhood $/2 of r such t h a t lh(t) - h(r)I < ellhlt for all h e g and all t • $/2- By the

equicont inui ty of {z* : t • T}, (11.7), we can shrink the ne ighborhood $/1 if necessary so

t ha t Izt(s) - zt(a)[ < e for all t • T and s • A[1. Let ~V = $/1 x $/2, and let (8, t) be an

44

arbi t rary point of A/. Let u be an arbi t rary point of B. Then, with u = ~n__ i g~ ® yi,

l ( ru) (s , t) - ( ru) (~ , ~)I

___ ICr~,)(~,t) - ( r ,~)(m t)l + t ( r u ) ( m t) - ( r , , ) (o , ~)1

= I J ( z s - u s ) C O - J ( z ~ , - u~,)(t)l + I J ( z ~ - u ~ , ) ( t ) - J ( z , , - -~)(~)1

< I I y ( z s - ~ ) - J ( z ~ - ,,~)11 + IIZ(z,, - u~) l l~

_< ~llCz,~ - ,~s) - (zo. - ~,~)ll + { ~ l l z ~ - uo.tl + t l J O l l } ~

_< ~t lz . , - ~,,I1 + ~ll'~.~ - ~o.II + ( ) , l t z t l + -~1t~tl + I I JO l [ } -c

_< + - + + + llZoll i = l

_< ~ + ~ k + .~l lzl l + ~k~ + IIJ011~.

Thus r(B) is an equicontinuous subset of V. •

2 .24 T H E O R E M . Let G and H be subspaces having linear proximity maps in

Banach spaces X and Y respectively. For any uniform crossnorm ~ on X® Y, the subspace

X b H + G b Y

also has a linear proximity map, and is therefore proximinM and complemented in X ®~ Y.

P R O O F . Suppose tha t P : X---~G and Q : Y---~H are linear proximity maps. Then

P ®a Iy and Ix ®a Q are linear proximity maps of X ®a Y onto G ~ Y and X ~ H,

respectively. One verifies readily that they commute with each other. Their Boolean sum

is therefore a projection of X ® ~ Y onto X ~ H + G ~ Yr. In order to see that the Boolean

sum of two commuting linear proximity maps (having a common domain) is again a linear

proximity map, write

[[I-A-B+ABH = I I ( Z - A ) ( I - B ) I I ~ 1. •

2.25 E X A M P L E . Let s i , . . . , s , be points in a compact Hausdorff space S, and

let G be the subspace (in fact an ideal) of C(S) consisting of functions which vanish on

{ s i , . . . , s , } . Similarly, let H be the ideal of functions in C(T) which vanish on { t i , . . . , t,~}.

Then C(S)®~ H+G®x C(T) has a linear proximity map. In order to employ the preceding

theorem, we have to show that G and H have linear proximity maps. A linear proximity

map for G is defined by

Px = z - ~ x(s~)ul i = 1

45

where { u l , . . . , u•} is a p a r t i t i o n of un i ty in C(S) sa t i s fy ing u,(sd) -= 6~j. For an a r b i t r a r y

g E G, the fol lowing inequa l i t y verifies t h a t P is a p r o x i m i t y m a p :

llx - Pxll = llEx(si)u, ll <_ maxEIx(s i ) [u i (s ) S

< m ~ lx(s,)t = m2x l~(s,) - ~(s,)l < IL~ - 91t-

If S = [a,b], one can take a = s l < s2 < " " < st, = b and cons t ruc t each u~ as a

piecewise l inear func t ion (f i rs t-degree sp l i n t funct ion) w i th kno t s a t s l , . . . , s,~.

For fu r the r i n f o r m a t i o n on subspaces which have l inear p r o x i m i t y m a p s , consul t [50,

125].

2 . 2 6 T H E O R E M . Let S and T be finite measure spaces, Let G and H be 6nite-

dimensional subspaces in L1 (S) and L1 (T), respectively. Then L1 (S) ® H + G ® L1 (T) is

proximinal in L I ( S × T).

P R O O F . The subspaces U = G ® L , ( T ) and V = L , ( S ) ® H are p r o x i m i n a l i n L , ( S × T ) ,

by 2.13 and 1.15. Note t h a t U + V is closed, by 11.2. Since V is p rox imina l , a p r o x i m i t y

m a p A : L I ( S × T) --* V exists . By 2.11, A has the p r o p e r t y t h a t for a lmos t a l l s, (Az)s

is a bes t a p p r o x i m a t i o n to Zs in H. By 2.21, the p r o x i m i n a l i t y of U + V wil l follow if we

can show t h a t for each z the m a p F(u) = A(z - u) is weak ly c o m p a c t f rom U to V.

Let B = {u e U : Ilull < k}. I t is to be proved t h a t F ( B ) lies in a w e a k l y c o m p a c t

subse t of V. Select a b i o r t h o n o r m a l sys t em { h , , ¢ , } ~ for g . Thus llhill = [[¢,ti = 1 a n d

= m r ( u ) . Then v e V, and for Y ~ i = 1 ¢~(y)h~ for each y E H. Now let u e B and v =

a lmos t a l l s, vs is a bes t a p p r o x i m a t i o n to zs - us. Hence

II,-'sll -< 2Jl~s - usll _< 2ll~sll + 21lusll.

Since Vs C H, we have v(s, t) = ~'~/m=l ¢,(vs)h , ( t ) , whence

(3) I,.,(s, t)l _< ll,-'sll ~ lh~(t)l = llvsllh(t) _< 2(llzsll + INsll)h(t). i = l

m Here we have p u t h = ~ = 1 lh~l • W i t h a s imi la r a r g u m e n t invo lv ing a b i o r t h o n o r m a l

sys t em for G we o b t a i n

iuCs, t) l _< IlutllgCs) u • U.

F r o m the las t i nequa l i t y we ob ta in , for u E B,

(4) llusll = IT lu(s , t ) ldt _< / T l ju'l lg(s)dt = g(s)llult _< k~,Cs).

Thus, from (3) and (4), we have

(5) lv ( s , t ) I < [21Izsll + 2kg(s)]h(t) = F(s , t ) .

46

Since the function on the right is an element of L1 (S x T) and is independent of u, we

have

lim / /E iV(s ' t ) ldsdt < lim / / E F ( s ' t ) d s d t = O

uniformly in v, as u ranges over B. By the Dunford Compactness Theorem, P(B) lies in

a weakly compact set. Note that the boundedness of r(B) is implied by (5). |

The following example, due to von Golitschek, shows how the approximat ion problems

considered in this chapter may be sensitive to alterations in the domain.

2 .27 E X A M P L E . Let S = T = [0, 0.3], and define the domain

D = { ( s , t ) : 0 < s / 2 < t < s < 0 . 3 } .

Let W be the subspace of C(S x T) consisting of functions

w(s, t) = x(s) + y(t) x • C(S), y • C(T).

By 2.22, W is proximinal in C(S x T). But if R denotes the restriction map from C(S x T)

to C(D), then R(W) is not proximinal in C(D); in fact, it is not closed. This can be seen

by considering the following functions in W :

~ . ( s , t) = x . ( s ) + y . (t) (~ = 3, 4 , . . . )

where yn(t)= -xrt(t) and

l o g l o g l / s if l /n < s < 0.3

x,(s) = log logn i f0 < s < 1/n.

One can prove the following facts.

(i) The sequence ]lwn]lD is bounded.

(ii) Every representat ion of Rw, has the form (xn - c,) + (y, + c,) for some constant

c,~. (The topological nature of D must be taken into account in proving this

equation.)

(iii) For any c,~ E JR,

I1~. - c.II + Ily. + c.II -+ oo.

By 11.2, R(W) is not closed.

47

C H A P T E R 3

T H E A L T E R N A T I N G A L G O R I T H M

Let X be a Banach space and U a subspace of X. Recall that a mapping A : X---oU

is a p r o x i m i t y map if

dist (x, U) = IIx - A ]J (x • x ) .

Thus, A x is a best approximation to x in U. It is clear that U is proximinal if and only if

it has a proximity map. Some subspaces have continuous or even linear proximity maps,

but these properties are exceptional. In Hilbert space, the orthogonal projection onto a

closed subspace is its proximity map.

Now suppose that two subspaces with proximity maps are given, say A : X---~U and

B : X - - o V . Is it possible to construct from A and B a proximity map for U-b V? It is

natural to try the following iterative process. Starting with xo, we compute xl = xo -

Bxo , x2 = x l - A x l , x3 = x 2 - B x 2 , and so forth. It is clear that IIz011 > fixlll > tlx21I ~ . . .

and that each x,~ differs from x0 by an element of U + V. The procedure just described

is called the A l t e r n a t i n g A l g o r i t h m . It originated with yon Neumann in 1933 [135].

When the circumstances are favorable, limn--.oox,~ exists, and x0 - lim~-~ooxn is a best

approximation of x0 in U + V.

A succinct description of the algorithm goes as follows. Let E = ( I - A } ( I - B) . Then

x2,~ = Er~xo . Observe that if the proximity maps A and B are linear, then E --- I - ( A $ B ) .

In this case, A and B are (bounded, linear) projections, and the theory of Boolean sums

(11.1) can be applied. Thus we have:

3.1 T H E O R E M . I f the prox imi ty maps A and B are linear and i f A B A = B A ,

then the al ternating algori thm produces a solution in two steps; i.e., xo - x2 is a best

approximat ion o f xo in U + V.

A basic result, established by yon Neumann, is that in Hilbert space the alternating

algorithm is effective for any pair of closed subspaces. This theorem is proved below

(Theorem 3.8).

A s m o o t h p o i n t in a Banach space X is a point x such that there exists a unique

functional ~ e X* satisfying ][~H = 1 and ~(x) = [Ix N. If every nonzero point of Z is a

smooth point, then X is said to be s m o o t h .

3.2 L E M M A . Let A : X- - - -U and B : X - - ~ V be prox imi t y m a p s on a Banach space

X , and let E = ( I - A) ( I - B) . Consider the fonowing propert ies o f a poin t x in X :

(i) x is a t~xed poin t o r E ; i.e., E x = x

(//) ll~ll = dist (x, U) = dist (x, V)

(iii) II~ll -- dist (x, U + Y) .

Then (i) :=~ Oi) always; (iii) =~ Oi) always; Oi) ::~ (i) i f U and V are Chebyshev subspaces;

and (ii) =¢, (iii) i f x is a smoo th point.

P R O O F . The equat ion x = E x r e d u c e s to x = x - B x - A ( x - B x ) and t o - B x =

A ( x - B x ) . This shows t h a t B x E U. Therefore

Ilxll -- llz - B x - A(x - Bz) i I = dist (x - B x , U) --- dist (z, U).

Then it follows tha t

ltxll = dist (z - B x , U) < ltz - Bxll = dist (z, V ) < Ilxll-

This proves tha t {i) =~ (ii). If (ii) is true, then by Singer 's Theorem [159, p. 18], there

exist funct ionals ¢ E U ± and ¢ E V ± such tha t I1¢11 = I1¢11 = I and Ilxll = ¢(~) = ¢ ( x )

If x is a smoo th point , then ¢ = ¢, and consequent ly ¢ E U ± N V ± = (U + V) ±. By

Singer 's Theorem, [[xi[ = dist (x, V + V). Thus (ii) =~ (iii) if x is a smoo th point . The

impl ica t ion (iii) =~ (ii) is trivial: [[z[[ = dist (x, V + Y) < dist {x, U) < ]]xll. Similarly,

Ilxll = dist ( z , V ) . If (ii) is true, then 0 is a best approx ima t ion of x in U. If U is a

Chebyshev subspace, then A x = O. Similarly B x = 0 if V is a Chebyshev subspace. Then

we see at once tha t E x = x. I

3.3 T H E O R E M . I f X is a smooth space, and f i E is contractive, then the sequence

{x2n} de~ned by the al ternating algorithm is convergent. /~ r the rmore , x0 - lim x2,~ is a

best approximat ion of Xo in U + V .

P R O O F . By Banach ' s Cont rac t ive Mapping Theorem, the m a p E has a unique fixed

point , x*, and the sequence x z , = E n x o converges to it. By 3.2, [Ix*It = dist (x*, U + V).

By induct ion we see easily tha t xo - x,~ E U + V for all n. Hence xo - x* E U + V. Then

llx0 - (xo - x*)ll = Ilx*tI = dist (x*, U + V)

= dist (x*, U + V) = dist (x* + (x0 - z * ) , ~ )

= dist (x0, U + V). I

The m o d u l u s o f c o n v e x i t y of a Banach space X is the funct ion ~ : (0, 2] ~ [0, 1]

defined by {1 } ~(e) = inf 1 - ~tlx + y l l Ilxll -< 1, Ilyll -< 1, I1:~ - yll >- ~ •

49

3 . 4 L E M M A . I f 0 < I1~11 < ½11x+Yl l t h e .

IlYll ~(11~ i YlI / I lYlI) < IlYll - I1~11.

P R O O F . We have

0 < II~II -< 511~ ÷ yll ~- II~II ÷ ~l lyl l ,

so II~II -~ llull. Let u ---- x/ l lyN, v = y/ l ly l l , a n d ~ = II~ - vll. T h e n by the d e f n i t i o n of 5,

1 6(~) _< i - 711~ + vii.

When we substitute the definitions of E, u a n d v the result is

llYll 5 ( l l x - ylI/IIyN) < l l y l l - ~II • + yll-< l l y l l - llxll •

A Banach space is said to be u n i f o r m l y c o n v e x if its modulus of convexity is a

strictly positive function.

3.5 L E M M A . If X is uniformly convex, then the iterates in the alternating algo-

rithm satisfy I l x . - x . + l N - ~ o

P R O O F . The conclusion is obvious if ][x~,[] ---, 0. We assume therefore that

l i m . +ooNx,,ll -- k > 0.

For appropriate points w,, in U U V, we have

1

= ~ I I ~ . + ( ~ . - w . ) l l

1

By 3.4, it follows that

IIx.II 5(t lx. - x . + l t l l l l x . l l _< t l x . l f - l lx.+~ll ,0.

Since I1~-11 -> k, ~e hav~

5(11~.-~.+~I1/11~.11) ,0.

By the uniform convexity, I1~ - ~.+~11 -~ o. •

50

A mapping F defined on a subset of a normed space is said to be n o n e x p a n s i v e if

[ [ r x - ry l l < [ I x - y[[.

It is said to be o d d if F ( - x ) = - F x .

The following theorem is due to Baillon, Bruck, and Reich [19].

8.6 T H E O R E M . Let C be a symmetr ic , dosed, convex set in a un i formly convex

Banach space. Let T : C --~ C be a m a p which is odd and nonexpansive. I f ~ is a point in

C for which l im,~(T '~ - Tn+l~) = O, then the sequence {T"~} converges to a f ixed point

o f T .

P R O O F . Since T is odd, T(0) = 0. Therefore,

IIT"~It -- IIT"~ - Toll < I IT"- I~ - oli = IIT"-I~II •

I t follows tha t the limit A -- l im, llT-~II exists. We assume tha t A > 0 because the other

case is trivial. For each i,

l im,~(T'~ - T"+ i~) = l i m [ ( T n ~ - Tn+I~) + . - . q- ( T " + ' - I ~ - T'~+~()] = 0.

Furthermore,

lIT"5 + T"+'SII = I 1 ~ - T " + ' ( - 5 ) I I < 1 1 ~ - 1 ~ - T " + ' - l ( - 5 ) I I

= I I T " - ~ , ~ + T " + ' - I ~ I I .

We have now 1 1

-< IIT"~II _< ~IIT"~ + T"+~I I + I I T " ~ - T"+'~II.

By taking the limit as n ~ oo, we get

_< lim,~ ~IIT",~ + A T"+i~[[.

Because of the monotonici ty established previously,

A < 1 _ ~IIT"~ + Tmeil (all n and m).

Let e ~-* ~(e) be the modulus of uniform convexity. We will show tha t the sequence

{T"~} has the Cauchy property. Let e > 0. Select 0 so tha t 0 < 0 < 1 and A / ( A + 0 ) >

1 -- 6(e/(A + 1)). Select an integer n such tha t [[T"~[ 1 <_ A + 8. Let i , ] >_ n.

Then

<1, <1,

A 6 + > > 1 - (~-=.). +-----~ A

51

F r o m the def in i t ion of the m o d u l u s of convexity, i t follows t h a t

< $ <

Hence ] I T i ~ - TJ~H < e, and the Cauchy p r o p e r t y is e s t ab l i shed . Since the space is

comple te , the l imi t z = l i m n T ' ~ exists . Since C is closed, z E C. Since T is con t inuous ,

T z = T ( l i n ~ T ' * ~ ) = l im T ~ + I ~ = z. •

3 . 7 T H E O R E M . I f X is smooth and uniformly convex, and f f

A : X - - - ~ U and B : X - - ~ V

are proximi ty maps such t h a t I - A and I - B are nonexpansive, then the alternating

algorithm is effective. Thus lim,~--.oo ( I - E n) is the proximi ty map for U + V.

P R O O F . I t is easy to see t h a t the m a p E . = ( I - - A ) ( I - B) satisfies the h y p o t h e s e s

in 3.6. By 3.6, {E'~x} converges to a fixed po in t of E . By 3.2, x - l i m E n x is a bes t

a p p r o x i m a t i o n of x in U + V. •

3 .8 T H E O R E M . In Hilbert space, the alternating algorithm is effective. Thus

l i m o o ( I - E '~)

is the orthogonal projection onto U + V.

P R O O F . The p r o x i m i t y m a p s A : X---~U and B : X - - ~ V are l inear and of n o r m 1 in

th is case. Hence I - A and I - B are l inear and of no rm 1. Now a p p l y 3.7. •

3 .9 I , E M M A . Let U and V be proximinal subspaces in a Banach space X , and let

{xn} be a sequence produced by the alternating algorithm starting at x. Assume that

[[xnl] _> k > 0 for all n. I f n is even select ~on E U ± and i f n is odd select ~,~ E V ± so that

11 ,,11 = I1 .11 and = II ,,ll =.

Then

8*(II . - .+lll/ll .tl) ,o,

where 6" is the modulus of convexity

P R O O F .

+ 2,,II >

>

of U ± + V ±.

1 ~(~ , ,+1 + ~,,)(:~,,)/ l lx~,~l l

( ~ , , + , ( ~ , , ) + It~,,t1~}/211~,~II

{~,,+,(~:~,.+~) + 11~:~11~}/211~,.II

{11~:~,,+,11 ~ + 11~,,ll~}/2tlx~.ll

11~,,+:,11 = I1~,,+~tl > o.

52

By 3.4,

I1~..11 ,~*(11~'~, , .+, . - w~ , , , . l l / l l~ . . t l ) -< l lw~, - , l l - I Iw~. .+ l l l .

A similar result holds t rue for 2n and 2n - 1. Hence for all n,

II~',.II ~ * ( I I ~ ' . + ~ - ~ ' - l l / l l ~ . , l l ) -< II~,.,II - I I~ , .+~I I

and

,~*(tt,,,:,..+1 - ~.11/11~..11) < I1~.11 - I1~..+111 , 0 . •

3 .10 T H E O R E M . Let U, V and U + V be closed subspaces in a Banach space

X. I f X and U ± + V ± are uniformly convex then the alternating method produces best

approximations in U + V.

P R O O F . Let {xn} be the sequence p roduced by the a l te rna t ing a lgor i thm, s ta r t ing wi th

x. For each n, there is an element 5o~ E X* such tha t lt~on[[ = Hx,~[[ and ~n(xn) = [[x~[[ 2.

Since x2k = x2k-1 - Ax2k-1 , and x2k+l = x2k - Bx2k, we can assume fur ther t ha t

5O2k E U ± and W2k+l E V -L.

Henceforth, w will denote an a rb i t ra ry element of W satisfying [[x - w[[ < [[x[[. Since

U + V is closed, there exists (by 11.3) a cons tant c such tha t w has a representa t ion

w = u + v , w i t h u E U , v E V , and HuH+HuH<el[w][ <2c[[x[[ .

Se lec t ~ ~ u and ~ E V so t h a t ~. + ~ . = x - ~ . and II~.H + l l~. l l < c t l ~ - ~2.11 <

2~II~11. T h e n

I1~=.11 ~ = ~ . ( x ~ . ) = ~ . ( ~ - ~ . - ~ . ) = ~ . ( ~ - ~,~)

= t l~ , , l l ll~ - wll + ~ ,2 , ,b , - ~ , , )

= I1~,,.11 I1~ - ~11 + (~,,.,. - ~ , , . - 1 ) ( , - , - ~,.,.)

<- I1~,.,.11 I 1 ~ - ,,,it + It~,.,. - ',,::'~,.-, 11 {ll'-'ll + I1~,--II}

-< I I~ r , II 11~ -- "-'11 + 4cl l~i l I1',o2,-, -- ~ , , - l l l - In this inequality, we now take an inf imum on w, get t ing

I1==,.,.11 = -< ;11~2,,.11 + 4,:11=11 I1~=,-,. - ~=,-,-111

where p = dist (x, U + V). If lim,~ I Ix . I I = 0 , w e a r e finished. In the o ther case, 3.9 shows

tha t I [ ~ - - ~ , - 1 [[--~ 0, and so we have lim [[x2,[[-- p. Hence lim [Ix2,+1][ = p. This means

tha t the points wn = x - x,~ form a minimizing sequence in U + V for approx ima t ing x.

Since X is uni formly convex, ]im w,~ exists and is a best approx ima t ion of x. •

The next few lemmas are needed to establish Theorem 3.16 concerning geometr ic

convergence of the a l ternat ing algori thm.

53

3 .11 L E M M A . I f A and B are linear proximity maps and the l imit x* = limn-~oo x,~

exists in the alternating algorithm, then 11~2.+2 - ~*ll <- 011~2. - ~*tl, w i t h 0 -- I I E I W l l

P R O O F . Since x* is a fixed p o i n t of E , and E is l inear ,

11~2.+2 - z*ll = I I E : : . - E~*II = IIE(z2~ - ~*11

_< IIEIWII I lx=. - =*11

Here i t is a lso necessary to note t h a t x 2 , - x* E W = U + V. •

11.12 D E F I N I T I O N . I f U and V are subspaces of a normed space X, we deiine

their inclination to be

incl ( V , V ) = s u p { ¢ ( ~ ) : ¢ e X * , . • U, ~ e V, II¢II = [lull = II~ll = ¢ ( - ) = 1}.

3 .13 L E M M A . I f U f? g = 0 and i f U + V is closed and uniformly convex, then

incl (U, V) < 1.

P R O O F . Assume the hypo theses and t h a t incl (U ,V) = 1. Then there exis t

¢n E X * , Un EU~ and Vn E V

such t h a t II¢,dl = 11~-11 = I1'~-tI = ¢ - b ' - ) = 1 and ¢,,(v,~) - + 1. S ince

1 + ¢ , , ( v . ) = ¢ . ( u . + vn) _< I1~- + ,,~11 <- 2

and s ince ¢,~(v,~) - * 1, we have [[u,. + v,,[[ -+ 2. Since V + V is u n i f o r m l y convex ,

Since U N V = 0, each e lement of U + V has a un ique r e p r e s e n t a t i o n as a sum of an

e l emen t of U and an e lement of IF. Since U + V is closed, there exis ts by 11.3 a c o n s t a n t

c such t h a t Ilu,,ll + ]lv,,H _< c][un - v,,[I. This is not poss ible , as ]]u,~[] = ]lv,~H = 1 and

I 1 . . - ~ . l l - ~ 0. I

A subspace U in a Banach space X is sa id to be s m o o t h if each nonzero e l emen t u

of U is a s m o o t h po in t of X.

3 .14 L E M M A . / f P is a n o r m - / projection of a Banach space X on to a smooth

subspace U, then for any other subspace V,

incl (U ,V) = [[PIV[I.

P R O O F . Since U is smoo th , there co r r e sponds to each u E U \ 0 a un ique func t iona l

¢~, E X* such t h a t 11¢~,II = 1 and ¢~,(u) = llull. By the H a h n - B a n a c h T h e o r e m , there

5 4

c o r r e s p o n d s to each x E X a func t iona l ¢~ E X* such t h a t IlCxll = 1, ¢~ ± k e r P and

Cx(x) = d is t (x, ker P ) . Since IIP][ = 1, I - P is a p r o x i m i t y m a p . I ts r ange is of course

k e r P . Hence ¢~(x) = I l x - ( I - P ) x H = I]Px][. I f u e V then ¢ , , (u ) = [[u[[, and so ¢~, = ¢,, .

Hence ¢ , E ( k e r R ) ± . Th is es tabl i shes t h a t ¢ , ( x - Px) = 0 for all u E V and al l x E X.

Now

IIPIVII= ~up IIP~ll = sup s . p ¢ ~ , ( P v ) = sup sup ¢~,(v). v E V v c V u E U v E V u E U

II~ll = 1 I1~11 = 1 I1-11 = 1 II~ll = 1 I1~11 = 1

The l a t t e r express ion is incl (U, V) since U is smoo th . I

3 .15 L E M M A . Suppose that, in the alternating algorithm, A and B are linear,

while W ( = U + V) is smooth and uniformly convex. I f (ker A n W ) + (ker B A W) is closed,

then IIEIWII < 1.

P R O O F . F i r s t we prove t h a t ker A N ker B (3 W = 0. If w is an e lement of th is set,

then Aw = B w = 0. Consequen t l y ][w[[ = d is t (w,U) = dis t (w ,V) . By 3.2, [[w]t =

d is t (w, U + V) = 0. Now by 3.13 and 3.14 we have

JlEIWII = sup I1 ( I - A)(I- B)~II ~ sup I I (X- A)~II w E W x E W A k e r B

II~ll = 1 Ilxll <_ 1

= i n c l [ k e r A N W , k e r B M W ] < 1. I

3 . 16 T H E O R E M . In the alternating algorithm, assume that the proximity maps

A and B are linear, while W ( = U + V) is smooth and uniformly convex. I f (ker A N

W) + (ker B N W) is closed, then the iterates x - xn converge geometrically to a best

approximation of x in W.

P R O O F . Use 3.7, 3.11, and 3.15. I

55

C H A P T E R 4

C E N T R A L P R O X I M I T Y M A P S

A proximity map A from a Banach space X onto a subspace U is said to be a central

proximity m a p if for all x E X and all u C U,

IIx - A x + -lI : I1:~ - A ~ - -t1.

Such maps are rare, but there are notable examples. The concept is due to Golomb [78].

He observed tha t the analysis of Diliberto and Straus depended only on the centrali ty

proper ty of the proximity maps. The Dil iberto-Straus-Golomb theory is presented next,

in a form modeled on that in [78]. We refer to the definition of the al ternating algori thm

as given at the beginning of Chapter 3.

4.1 L E M M A . Ill the prox imi t y maps A z~nd B are central, then the iterates in the

al ternat ing algori thm have these properties:

(0 IIx.I[ = II~- - 2~.+1]1

(/0 u ¢ ~ x * and I1¢1I < 1, then ¢(x.) >_ :Z¢(~ .+1) - I I ~ . I I -

P R O O F . If n is odd, then (because A is a central proximity map)

112x.+1 - x . l l = l ix-+~ + ( ~ - + ~ - ~-)Ii = 11~- - A ~ . - A * . l l = 11~- - A:r,~ + A x . l l = fix-li-

The proof for even n is the same, except tha t B replaces A. Now if II¢ll < 1, then

This yields (ii) immediately. |

4.2 L E M M A . Let A : X ..... ~ U be a central proxJmi ty map . Corresponding to each

e x and ¢ ~ X* there is a~ element ¢ e X ~" such that II¢lI = II¢lI, ¢ + ¢ ~ v ± , and

¢ ( x - Ax) = ¢ ( x - Ax) .

P R O O F . Define ¢ on the subspace M generated by U and x - A x by put t ing

¢ ( u + X ( x - / i x ) ) : ¢ ( - u + ~ ( x - A x ) ) .

Here, u E U and A E 1/.. I t is clear tha t (¢ + ¢ ) (u ) = 0 and tha t ¢ ( x - Ax) = ¢ ( x - Ax).

Since A is a central proximity map, we have the following equat ion (in which A ~ 0 and

~, = ~ /~ )

I¢(u + a ( z - A x ) ) l = I ¢ ( - u + A(~ - Ax))I = I~1 I ¢ ( - ¢ + • - Ax)l

< I~1 I1¢I1 I I - u ' + z - Axll = [A I I1¢11 I lu '+ z - AxII = ll¢ll I l u + A(z - Ax)[ I. Thus the no rm of ¢ on M does not exceed 11¢tl. An appl icat ion of the Hahn -Banach

Theorem completes the proof. •

4 .3 T H E O R E M . Let U and V be subspaces of a Banach space hav ing central

proximity m a p s A and B, respectively. If U + V is dosed, then the sequence {x,~} generated

by the alternating algorithm has the property ]]x,~}j ~ dist (x, U + V).

P R O O F . Fix two even integers m and k. Let n = m + k. By the Hahn-Banach theorem

there exists a funct ional ¢0 e S ± such tha t II¢oll = I and ¢ 0 ( x , ) = Ilx.ll . By 4.2, we can

define induct ively the funct ionals ¢1, ¢2, - - - , Ck so tha t

(1) ¢ i + ¢ i - l E U ± f f i i s o d d , 1 < i < k

(2) ¢i + ¢i--1 E V / if i is even, 1 < i < k

(3) ll¢,ll -< 1, 0 < i < k

(4) ¢i (z ,~ - i+ l ) = ¢ i - l ( xn - i+ l ) , 1 < i < k.

We now assert t ha t for each r in the range 1 < r < k,

(5) ¢ , ( x . _ r ) > l l xm l l - 2 r ( J lxml l - IIx.II), 0 < i < r.

This will be proved by induct ion on r. For r = 1 the proof is conta ined in the following

inequality, based upon the fact tha t ¢0 and ¢1 belong to U±:

¢ 1 ( x , ~ - 1 ) = ¢ 1 ( x ~ + A z . _ ~ ) = ¢~(x,~) = ¢0(z ,~) = ¢ 0 ( x , ~ - ~ )

= [Iz.II __ Ilz.II + (llx.II - II~mll) -- I I~ml l - 2(11~. ,11- IIx.ll). We n o w a s s u m e the v a l i d i t y of (5) for r = t < k, and prove its v a l i d i t y for r = t + 1. B y

4.1,

>_ 2 ¢ d ~ . - d - l l ~ m l l

>_ 2{Jl~mll - 2 ~ ( l l ~ l l - t t~ . l l )} - t I~II

= I l z m l l - 2 ~ + ~ ( t l z m l I - I1~-II). T h u s , the i n e q u a l i t y (5) is true for r = t + 1 = i . If 0 < i < t, then as before,

¢ , ( x n - t - 1 ) >_ 2 ¢ , ( z . _ t ) - lIx,~_t_llI

-> z{ll~:,,,ll - 2~(l lzmll- l lz.ll)} - llxmll - - [ J x m [ l - 2~+i(II~,,,II- [ I z . l l ) .

57

Thus (5) is fully established for 1 < r < k.

Now observe that ¢0, ¢ 1 , . . . , ¢k all belong to U ± + V ±. Since U + V is closed, U ± + V ±

is also closed, by 11.3. Hence, by 11.3, there exists a constant c such that each functional

¢ in U ± + V ± has a representat ion of the form ¢ = ~ + ¢, with ~ E U ±, ¢ E V ±, and

[[0[[ + [[¢[[ < c[[¢[]. We apply this to ¢k, getting ¢k = 0 + %b and [[0][ + [[¢I[ -< c. Define

¢k+1 = - -¢ - 4o. Obviously 11¢k+i 11 _< c + 1 and 4k+1 + 4o = - ¢ 6 V ±. Also

tk+x + tk = - ¢ - ¢ o + e + ¢ = e - ¢0 ~ U ±.

v ' k + l U ± V ± Define ¢ = z-.i=o ¢i. Then ¢ E N because

¢ = (¢0 + ¢1) + (¢2 + ¢~) + + (¢k + tk+~) e c ±

and

¢ = (¢, + ¢2) + - - . + (¢k-1 + Ck) + (¢k+i + ¢o) e v ±.

Since I1¢11 < k + 1 + c + 1 ---- c', we have

dist (~, ~ + V) > ¢ (~) /~ ' = ¢ ( ~ ) / ~ '

= (c') -1 ¢ , ( ~ ) + ¢k÷1(~. , )

k + c + 1 > c, X{ll~mll- 2k{ll~"~ll -lt~.~÷kll)} - - 7 - J J ~ l l

Now let p = lim Hx,~ll. In the previous inequality, let k be fixed while rn --~ oo. The n - - ~ OO

result is k - - c

dist (x, U + V) >_ k + c + 2P.

Now let k--* oo to obtain dist (z, U + V) >_ p. •

4.4 T H E O R E M . I f X is uniformly convex, i f U + V is closed, and i f A and B are

central proximi ty maps, then the alternating algorithm produces a sequence {x,~} such

tha t l im (x - x , ) is the best approximation of x in U + V. B ~ O 0

P R O O F . By 4.3, x - x ~ E U + V and [[x, II ~ dist ( x , U + V ) . Hence { x - x , ~ } is a

minimizing sequence in U + V for x. By the uniform convexity, ~imoo(~ - ~ , ) exists and

is the best approximat ion of x. •

4.5 L E M M A . Let A : X ~ U be a proximity m a p such that

(i) llx - 2axl l < tlx]l (x e X)

(i0 A(~ + ~) = A~ + ~ (~ e X, ~ e U ) .

Then A is a central proximi ty map.

58

P R O O F . I f u E U a n d x E X t h e n b y (i) and (ii),

I1. - A . + =11 -> I1(* - A . + , , ) - 2 A ( x - A x + =)11

= t1. - A x + u - 2 A x + 2 A x - 2u]]

= f i x - A x - uIl.

Now repeat this a rgumen t wi th - u in place of u to conclude t ha t

llx - A x + =11 : I1- - A x - ul]. •

4.6 T H E O R E M . E v e r y o r t h o g o n a l p ro j ec t i on on H i lbe r t space is a cen t ra l p r o x i m -

i t y m a p .

P R O O F . Since x - A x A_ A x , we have by the Py thagoras Theorem

l l~l l ~ : t l~ - Az + A x t ? : l l~ - A x l i ~ + t lA~t t ~ : TIx - A ~ - A ~ l t ~ = t l~ - 2 A ~ N ~-

Now use 4.5. •

Notice tha t 4.6 and 4.4 together give another proof of von N e u m a n n ' s Theorem, 3.8.

4 .7 T H E O R E M . A s u b s p a c e in a B a n a c h space can have at most one cen t ra l p rox -

i m i t y map.

P R O O F . Let A1 and A2 be centra l proximi ty maps of a Banach space X onto a subspace

U. F ix x E X and put u = A l x - A 2 x . Consider the equal i ty

This equal i ty is clearly t rue for k = O. If it is t rue for k = n, then

II * - A l x l l = II x - A , x + null = l[ :~ - A 2 z + (n + 1),,II

= i l x - A 2 x - (n + 1)u H = I l x - A l X - (n + 2)u l l - - I l x - A l X + (n + 2 )u H.

Thus , by i nduc t ion we have

Nx - Alxi[ = 11* - A l z + 2 n u H > 2n [lull - Nx - A l x H.

Let t ing n -* co, we see t ha t Ilull = 0. |

59

C H A P T E R 5

T H E D I L I B E R T O - S T R A U S A L G O R I T H M I N C(S × T)

In this chapter, a part icular instance of the al ternat ing algori thm will be considered

in detail. The setting is the space C(S x T) of continuous functions on S x T, where S

and T are compact Hausdorff spaces. The two subspaces which figure in the al ternating

algori thm are C(S) and C(T). Here we identify an element u e C(S) with an element

• C(S x T) by writing ~(s, t) = u(s). Henceforth we do not belabor this distinction.

The first investigation of the al ternating algori thm in this case was carried out by

Diliberto and Straus in [56]. Their work was independent of von Neumann 's , and the

results and methods are quite different; only the algorithm itself is the same. Thus it

seems appropr ia te to refer to the algorithm by the names of Diliberto and Straus. The

question of convergence of the algorithm was left open until the work of Aumann [6].

The simplest and most natural proximity maps of C(S x T) onto C(S) and C(T) are

given by l m~nz(8, t) (Az)(8) = ½ max z(s, t) +

1 (Bz)(t) = ½ msaxz(s , t ) +

With these maps in hand, one can then define a sequence {z~}, s tar t ing with any z0 •

C(S x T), by the formulae

z2n = z2n-1 -- A z 2 , - i

Z 2 n + l = Z 2 n - - Sz2..

Diliberto and Straus proved, among other things, that I[z,[[ ~ dist (z, C(S) + C(T)) and

tha t the sequence {z,} has cluster points. Aumann subsequently proved tha t the sequence

{z - z , } converges to a best approximat ion of z in C(S) + C(T).

It will be convenient to define an "averaging functional" 5{ on univariate functions

by writing

inf f ( s ) . = s u p + 8

We can then express A and B as follows:

( a z ) ( s ) = ~ tzs and ( B z } ( t ) = )az t

where Zs and z t are the "sections" defined by zs(t) = zt(s) = z(s, t).

5.1 L E M M A .

maps .

PROOF.

Ot ~ ~ X ~ o r

The m a p s A and B jus t defined are non-expansive, central, p rox imi t y

For a funct ion z E C(S) , the constant which best approximates x is given by

= ½ m ~ x ( s ) + ½ ~nxCs) .

Therefore, the opera to r B has the p roper ty tha t for arty y E C ( T ) and for any fixed t,

s u p Iz(s, t) - ( B z ) ( t ) l < s u p lz(s, t) - vCt)t . 8 S

Consequent ly

Ihz - B z l l < l l z - ylL.

This shows t h a t B is a proximi ty m a p of C ( S × T) onto C ( T ) .

It is apparen t tha t B is order-preserving and tha t B ( z + r) -- B z + r for any cons tant

r. Hence if r -- IIz - w]] then f rom the pointwise inequali ty

we conclude tha t

whence

- r + w < z < r + w

- r + B w <_ B z <_ r + B w

] ]Bz - Bw[ I ~ r = IIz - wl l .

This shows tha t B is non-expansive. In order to prove tha t B has the centra l i ty proper ty ,

it suffices to show t h a t for a rb i t ra ry z E C ( S × T) and u E C(T) ,

IIz - B z - ull > IIz - B z + ull.

Let w = z - B z , and select (so, to) so tha t

I1'-,' + ,,11 = o[(~ + ~)(so, to)] ~ - - ± 1 .

Since B w = O, there is a point sl such t h a t

~[~(so, to) + ~(8, , to)] _< 0.

Indeed, let 81 be a m a x i m u m point of w(s, to) if a < O, and let 81 be a m in imum point if

a > 0. Now we have

II w - ull -> ~[u(to) - wCsl , to) ] >_ ~[u(to) + W(so,to)] = I1 ~ + ull- U

61

5.2 L E M M A . The averaging funct ional ~ is non-expansive:

I ) [ f l - ) t f 2 l _< [if1 - f2 l l .

P R O O F . If a = I l L - f2ll then

f2-o~< fl <_f2+~.

By using obvious propert ies of ~ , we conclude tha t

JMf2 - c~ < At f l < .M f2 + 0~.

This implies the inequali ty to be proved. •

The a l te rna t ing a lgor i thm in the present set t ing will be te rmed the "Di l iber to-Straus

Algor i thm." Using the operators A and B described above, we write it in the following

form: { zo~C(SxT) z,~+1 =z ,~-w,~

wn = Azn if n is odd,

w , = Bz,~ if n is even. We will prove below tha t the sequence z0 - z,~ converges uni formly to a best approx ima t ion

of zo in C ( S ) + C ( T ) .

5.3 L E M M A . For two continuous functions, f and g, on a a compact domain,

sup f - s u p g _< s u p ( f - g).

I f equal i ty occurs here, then there exists a single point where all three suprema are at tained

simultaneously.

P R O O F . For each s we have

(1) / ( s ) = f ( s ) -- g(s) + g(s) <_ s u p ( f -- g) + g(s) < s u p ( f -- g) + s u p g

(2) fCs) = f ( s ) -- g(s) + g(s) <_ f ( s ) - g(s) + sup g _< s u p ( f - - g) + sup g.

Ei ther of these yields the inequali ty

sup f _< s u p ( f - g) + sup g.

Suppose now tha t equali ty holds in this inequality. Let ( be a point such t h a t f ( ( ) = sup f .

From (1),

sup(f - g) + supg ---- sup f = f(~¢) ___ sup(f - g) + g(~).

Hence g(~) = sup g. From (2),

s u p ( f - g) + s u p g = sup f = f ( ( ) _< f(~) - g(() + sup g.

Hence f (~) - g(~) = s u p ( f - g). •

We come now to one of the crucial lemmas established by Aumann . Having fixed z0,

we define the sequence

A . = max{ lz ,~(s , t ) [ : Iw.(s,t)l = I1~.11}-

62

5 .4 L E M M A . UlI~-II = 11"--~1l th~n ,~,_~ _~ ,~, + II"-ll-

P R O O F . Suppose that n is even; the other case is similar. Since B is non-expansive

and B z ~ _ 1 = O, we have

II,-,,.1I = IlBz.tl = ltBz,,- Bz,~-ltI <_ llz,~- ~.-~11 = II,,,.-~11 = I1~.II.

By the definition of )~n, there exists a point r E T such that

I~,..(T)I = I!~,.11 ~nd maxtz,~(8, r)l = ~,,. 8

There are now two cases, depending on the sign of w , ( r ) . We assume that w,~(r) = Ilw,~ll.

Since w,~ = B z , ~ - B z ~ - l , we have

½l%ax z,,(~, .,-) + ~ z , (.~, .,-)] - ~ 1 % ~ ~,.,_~ (.~, .r) + m~n ~,,_,. (~, ~-)] = Jl'"--~ il-

This can be rewritten as follows:

The bracketed expressions in the preceding equation do not exceed l iT- -1 ]1, as we see

with the help of 5.3:

m ~ . ( ~ , ~ ~) - m ~ . _ l ( ~ , ~) _< max[~ . (8 , ~) - ~--1C~, ~)]

= % ~ [ - , , , , . - 1 ( ~ , .,-)] _< 11,.,,,,-111.

The other term is analyzed similarly. We can now conclude that

m~x~.(~,~ ~) - % a x z . _ ~ ( ~ , ~) = II~--~11

%ax[z,,(..~, .,-) - z,_~(.% .,-)] = I1~,.-, II

m ~ [ - ~ , _ , ( . . , , , ,-)] = I1~,,.-111. 8

By the second half of 5.3, we conclude that there exists a point a such that

z , ( a , r ) = m a x z , { 8 , r) 8

~ ._~(~ , ¢} = ~ a x ~ . _ ~ ( ~ , ¢). 8

By previous equations,

0 < 119.11 = ~ . ( ~ ) = CBz.)C~) = ½ % ~ . ( ~ , ~ ) + ½ ~ n ~ C s , ~).

63

Consequent ly ,

and

Using the equat ion

m a x z n ( s , r) >_ - minzn(s, r) 8 S

)~,~ = m a x l z . ( s , r)[ = maxz,~(S,s r) = z,~(c% r).

1 mtaxzn_i(cr, t) -I- lmitnzn-l(cr, t ) ~._~(o) = ( A z . _ ~ ) ( ~ ) =

we now infer t ha t A . - 1 __> -- m i n z , , _ l ( a , t )

t

= m a x z . - ~ ( m t ) - 2 w . _ ~ ( ~ ) t

_> z . _ ~ ( ~ , d - ~ . - i ( ~ ) - ~ . - 1 ( ~ )

= z~(~, ~) + IIw.-~LI

= A n -~ [ [Wn- - i [ 1.

This completes the proof in the case w,~(r) -- Ilw, ll. In the o ther c~se, we write

- -~ . (~ ) = ( B ~ . _ l - B z , d ( d

and proceed as before. We find a point a such tha t

z ._~(~, ~) = rain ~ . _ l ( s , d , 8

and

zn(cr, r) = minz,~(s, r). 8

Then ~ - 1 >_ mpxz._l (~, t) ~ - z . (~ , d ÷ I1~'--1 II

>--M ÷ IIw.-lt[. I

For the remaining par ts of the analysis, we int roduce some convenient nota t ion:

Vn ~ WO ~ W2 ~ W4 ~ " '" @ W2n

un = Wl -t- w3 + w5 + "'" + w 2 n - i .

Then it follows tha t v~ E C ( T ) , un E C ( S ) , z~ . = z - u ~ - v . _ l , and z2,~+i = z - u . - v ~ .

64

Define opera tors A s : V -+ U and B ' : U ~ V by pu t t ing

A ' v = A ( z - v), B ' u = B ( z - u)

where z is the fixed member of C ( S x T ) which is the s ta r t ing point of the Di l iber to-Straus

i terat ion. The propert ies of A give us

0 = a (z2 ,~- I - az2r , -1 ) = az2,~ = A ( z - u,~ - v,~-l) = a ( z - v r , - 1 ) - u,~.

Hence un = A ( z - v n - 1 ) = A % , ~ _ , . Similarly, v,~ = B~u,~. Hence ur~+l = A%,~ = A ' B ' u n

and v,~+l = B ' u , ~ + , = B e A % , ~ .

5.5 L E M M A . F o r al l n,

lu. . ( .~) - u , . (o - ) l _< Ilz.~ - zo.II

I , . , , ( t ) - ,,~C',-)l ~ EI~ ~ - z~l l

Iz,( . . , , t ) - z , . (o- , , - ) l ~ 211z.~ - ~o.II + 211 z~ - z " l l .

P R O O F . As noted above, u,~ = A ( z - v , ~ - l ) . W i t h the help of the averaging funct ional

.M in t roduced previously, we have

l u , ( s ) - u,~(,~)l = I[ACz - v,-1)]Cs) - [ A ( z - vn_l)](c~)t

= l t z ~ - z ~ I I .

The proof for v,~ is similar. As for z~, , we have z2,~ = z - u n - v n - l , whence

< Iz(.~, t ) - ~(,:,, t ) l + I~.,.(.~) - u,.,Co-)l + Iz(,:,, ~:) - z(o-,- ,-) l + I , . . , , -1 ( t ) - v , - 1 ( . , - ) l

211z.~ - zo-II + 211z' - ~"11.

The proof for z2~+1 is similar. |

The next result is the second of the crucial lemmas due to Aumann .

5 .6 L E M M A . In t h e D i l i b e r t o - S t r a u s a l g o r i t h m , w e h a v e limwr, --- 0.

P R O O F . We have seen tha t ltwr~+llt <_ IIwni[ for all n. (See the first equa t ion in the

p roof of 5.4.) Hence we m a y define e = lim IIw, II. Because of equicont inui ty of {z,,} (5.5)

there is a un i formly convergent subsequence, z,~ --+ z*. If the a lgor i thm is applied to

z* as s ta r t ing point , the result is a sequence which we denote by z~, z ~ , . . . . Since the

opera t ions in the a lgor i thm are continuous, we have z * = l i m k ( z ~ , ~ + m ) . It follows tha t

[[Zm+ 1 -- z*nl [ = l iml]znk+m+l - Znl~+m] I : limllw,~k+m[ [ = e. By 5.4, applied to z*~, we

have A~ _< A * _ ~ - e for m---- 1, 2 , . . . . This leads to 0 _< A~ _< A 7 - ( r n - 1)e, and therefore

to the conclusion tha t e---- 0. |

65

5.7 L E M M A . The sequence {un} is bounded.

P R O O F . By 5.5 the sequence {zn} is equicont inuous . I t is c e r t a in ly b o u n d e d , s ince

llzoll --- tlzllt >- - . By the Ascol i Theorem, the sequence {z2n} has a convergen t subse-

quence, {z2n~,}. Since W is c losed and z,~ - z E W, we can wr i t e l imz2nk = z - u - v for

a p p r o p r i a t e u e C(S) and v e C(T) . Since Bz2n = 0, we have

o = B ( z - , , - v ) = B ( z - , , ) - v = B ' , , - ,~

Thus v = B'u . Since z~+ l - z n --* 0 by 5.6, we have z2n~,+l "* z - u - v . F r o m the e q u a t i o n

Az2,~+l = 0 we conc lude as above t h a t u = A'v. Hence u = A ' B ' u . T h e b o u n d e d n e s s of

{ u , } now follows f rom the non-expans iveness of AtBt:

]]u,~+l - ul] = [[A'B'u,~ - A'B'u[[ < [[u~ - ul]. |

5 .8 T H E O R E M . The sequence {z,~} produced from an arbitrary z E C ( S × T) by

the Diliberto-Straus Algori thm converges uniformly~ and z - l i ra zn is a bes t approximation

of z in C(S ) + C(T) .

P R O O F . By 5.5, the sequence {un} is equicont inuous . By 5.7 i t is b o u n d e d . Hence

by the Ascol i T h e o r e m there exis ts a convergent subsequence, {u~, k }. P u t u* = limk u , k.

T h e n (by an equa t ion jus t p r io r to 5.5) u,~k+~ = A'B 'unk --~ A 'B 'u* . By 5.6,

II~-+~ - ~ - l l = IIw=~+xll - ~ 0 .

Therefore Unk+l --* u* and A'B 'u* = u*. As in 5.7, we have

I1~,+1 - ~*11 = I I A ' B ' ~ . - A ' B % * I I _< I1~- - ~*[I-

This shows t h a t u,~ ---* u*. Hence v~, = B'u, , ---* n 'u* =. v*. By 4.3, [lzz,~H ~ d is t ( z , W ) .

Hence

and

I [ z - ~ . - v . - ~ l l ~ d i s t ( z , W )

IIz - ~* - v*ll = d i s t ( z , W ) . II

66

C H A P T E R 6

T H E A L G O R I T H M O F V O N G O L I T S C H E K

In a recent series of papers [69-74], yon Goli tschek has developed a powerful new

a lgor i thm for ob ta in ing sup-norm approx imat ions of the fo rm

(1) z(s, t) ~-, f[x(s)h(t) + y(t)g(s)]

in which g, h, f , z are all prescribed continuous functions, and the funct ions x and y are

sought . His a lgor i thm provides a const ruct ive proof of the following theorem:

6 .1 T H E O R E M . Lee S and T be compact Hausdorff spaces. Let

z E C ( S x T ) , g E C ( S ) , h E C ( T ) , g > 0 , and h > 0 .

Let f be a strictly increasing element of C(J~:~) such that f -1 C C(/R). Then z has a best

approximation of the form (1), with x e C(S) and y e C(T).

In fact, yon Goli tschek proves 6.1 for z E C(D), where D is a subset of S x T subject

to some technical hypotheses. Also he proves 6.1 wi th somewha t less restr ict ive condi t ions

on f. We prove Theorem 6.1 below.

In w h a t follows, z, g, h, f r emain fixed, and we set

W = { f o ( x h + y g ) : x e C ( S ) , y e C ( T ) } .

Observe tha t W is not a l inear subspace. However, since f is invertible, some techniques

of l inear app rox ima t ion theory are applicable.

The yon Goli tschek Algor i thm contains a real pa rame te r a chosen to lie in the interval

0 < a < llz - f o 0ll. If ~ > d i s t ( z ,W) , then in a finite number of steps the a lgor i thm will

produce an element w • W satisfying II z - wll _< a.

Having fixed ~, we define

t) = f - l [ z C s , t) - ]/gCs)hCt)

K(s, t) = f - l [z (s , t) + a]/g(s)h(t).

The a lgor i thm s ta r t s by defining

Xo(s) = 0

At the i th step, we define

and

and yo(t) = inf K(s , t).

x,(8) = x ,_ l ( s )v sup[k(s, t ) - yi_l(t)l t

yi(t) = y i - l ( t ) Ai~f[K(8 , t) -- xi(s)] .

If Yi = Yi-1, then STOP. (The symbols v and A denote the pointwise m a x i m u m and

m i n i m u m opera t ions , respect ively.)

6 .2 L E M M A . / f t i is a point such that y~(t~) <_ y i - l ( t i ) , then there exist points si

and t~-i such that

(i) y , ( t , ) = y , _ ~ ( t , _ ~ ) + K ( s ~ , t , ) - k ( 8 , , t , _ ~ )

(ii) y i - l ( t i - 1 ) < y i -~( t i -1 ) .

P R O O F . Let si be a m i n i m u m point for the funct ion K ( • , t i ) - xi. Let t i -1 be a

m a x i m u m point for the funct ion k(sl , • ) - Yi-1. Then

K ( s , , ti) - x, (8,) = in f [K(s , t ,) - x,(s)]

= y , ( t , ) < y , _~ ( t , )

<<_ g ( s i , t , ) - x , - l ( s i ) .

Hence, x , - l ( s , ) < x,(8,) . Similarly, we have

k ( s , , t , - O - y , - l ( t , - l ) - - s u p i k ( s , , t ) - y , - l ( t ) ] t

= ~,(8 , ) > ~ , - 1 ( s , )

Hence, y i - l ( t , - l ) < Yi -2 ( t , -1 ) . This verifies pa r t (ii). The verif icat ion of (i) is now as

follows: y,(t ,) = K(8, , t,) - x,(8,)

= K ( s i , t i ) - k ( s i , t i - 1 ) + Y i - l ( t i - 1 ) . •

6.3 L E M M A . / f n > 1 and ift,~ is a point such that y,~(t,~) < y,~-l(t,~) then there

exists a "path"

(81,t0) , (81,t1), (82 , t l ) , ( 8 2 , t 2 ) , ' ' ' , (Sn, tn)

such that

P R O O F .

n y . ( t . ) = ~-~]{n(8,, t , ) - k(8 , , t , _ l ) } + y0(t0)

Use the preceding l e m m a n t imes. •

68

6.4 D E F I N I T I O N . A path is an ordered set o f points

(sl,to), (s~,tl), (s:,tl), (s2,t~),. . , (s. , t ._l) , (~.,t.).

The pa th is said to be closed i f tr~ = to.

6.5 T H E O R E M . /if the algori thm s tops in the n t h step, then a >_ dist(z, W) and

[Iz - f o (ghx,~ + ghy~)[[ _< a. L¢a > dist(z, W ) then the algori thm will s top at some step.

P R O O F . The definition of yi yields the following pointwise inequali ty:

f - -1 0 (Z 71- OL) yi _< K - x~ = - xi.

gh

W h e n this is rearranged, it reads

- a < z - f o (x~gh + yigh).

Similarly, s ta r t ing with the definition of x~, we obta in

z - f o (x igh + y i - l g h ) <_ a.

It follows t h a t in the case y~ = Yn-1 we will have

- -a <_ z - - f ° (x,~gh + y , gh) <_ a.

This shows tha t Hz - f o (x,~gh + y,~gh)]] <_ a and tha t a _> p ~ dist(z, W). Now we want

to prove tha t if a _> p, then for some n, y,, = Yn-1. If this conclusion is false, then for an

a rb i t ra ry integer n there exists a point tn E T such tha t yr,(tn) < y u - l ( t , ) . Then by 6.3,

there exists a pa th

(~, to), (sl, tl), (~ , tl), (s~, t~), . . . , (s,,, t,,)

such tha t rt

Y,~Ctn) - yo(to) = E { K ( s , , t,) - k(s , , t , -1 )} . i = l

Select fl so tha t p < fl < a. By compactness and continuity,

/ - 1 ° ( z + a ) f - l ° ( z + / ~ ) >_c

gh gh

and f-~ o (~ -/~)

gh gh f-I o (z - ~) >_ c

69

for some e > 0. Since/~ > p, there exist x E C ( S ) and y E C ( T ) such that

Ilz - f o (~gh + ygh)ll < ~.

This leads to the inequality

and then to

- f l < f o (xgh + ygh) - z < fl

< x + y < gh gh

It follows that

f - ~ o ( z - : ) f -~ o (z + ~)

f - -1 o (Z -- (2) f - -1 o (Z + (2) k + e = + e < x + y < - e = K - e .

gh gh

From a previous equation, therefore,

n

y . ( t . ) - y o ( t 0 ) >_ ~ ' { [ x ( ~ , ) + y ( t , - 1 ) + El - i x ( s , ) + yCt,_~) - El} i=1

= y ( t . ) - yCt0) + 2 , ~ - ~ ~ as ~ - ~ oo .

This however is a contradiction, since Yo _> Yl -> "'" directly from the definition of the

algorithm. •

6.6 L E M M A . Let r e C ( S ×T). Define F - ( s ) = i n f t f ( s , t) and r + (s) = s u p t r ( s , t).

Then [or s, a E S we have

t F - ( s ) - F - ( a ) i <- NFs - Fall and IF+(s) - F+(a)[ < ItFs - Fall

where Fs is the s -sec t ion of F, de~ned by the equation Fs(t) = F(s , t).

P R O O F . For every s ,a , and t~

F - ( s ) g F ( s , t ) = F ( s , t ) - F (a , t ) + F(cr, t) < IlFs - F~,l] + F ( a , t ) .

By taking an infimum we get

F - ( s ) << HFs - Fall + F - ( a ) .

This establishes that

F - ( s ) - F - ( a ) < IIFs - Fall.

By interchanging s and or, we get

F - ( a ) - F - ( s ) <_ I I F , - Fsll.

7O

The las t two inequa l i t i e s i m p l y the one in the l e m m a . The inequa l i t y for F + is proved in

the s ame way. •

a n d

6 .7 L E M M A . For each n, s, or, t, r we have

lx . .C.q - ~ . . . (~)1 ~ IIk.~ - ko.ll

ly=Ct) - y . ( ~ ) l < I l K * - K~II.

P R O O F . Since x0(s) = 0, the first i nequa l i ty is t r i v i a l when n = 0. Since

yo(t) = inf K ( s , t)

the second inequa l i t y follows f rom 6.6 when n = 0. If the two des i red inequa l i t i e s are

a s sumed t rue for an index n, then for the index n + 1 we have

x ~ + , (~) = x , ( ~ ) v supIk(~, t) - y~(t)] . t

By 6.6 (spec ia l ized to a two-po in t set!) we see t h a t we need on ly verify these two inequa l -

i t ies:

(2) I,~,,.(~) - ~ . (o - )1 _< I Iks - ko.II

(3) I sup[k(s , t) - y . ( t ) t - s n p i k ( o , t) - y . ( t ) ] t < Ilks - k ~ l l t t

T h e first of these is our i nduc t i on hypo thes i s .

Indeed, the left s ide of (3) does no t exceed

ll(k - y . ) ~ - (k - y . ) ~ t l = IIk~ - k~It.

The second inequa l i t y in the l emma , for the case n + 1, is proved in the same way.

6 .8 L F , , M M A . I r a > d i s t ( z , W ) , then for each closed path,

P R O O F .

The second is e s t ab l i shed by us ing 6.6.

r t

~ [ K ( s , , t , ) - k ( s , , t , _ l ) ] > O. i=O

We prove the l e m m a unde r the a s s u m p t i o n t h a t ct > p = d i s t ( z , W ) . The case

a = p wil l t hen follow by con t inu i ty of K and k as func t ions of a . If a > p, then there

71

exist x e C ( S ) and y • C ( T ) such that llz - f o (xgh + ygh) ll <- a. This is equivalent to

each of the following inequalities:

-o~ < f o (xgh + ygh) - z < ot

z - a < f o ( z g h + y g h ) < z + a

f - 1 o (z - ~) < x~h + ygh <_ f - 1 o (z + ~)

[y-1 o (z -- a ) ] /gh < x + y < [ f - 1 o (z + a)J /gh

k < x + y < _ K .

Now define two linear functionals on C ( S × T) which are sums of point functionals

¢ = E ( s i , t , ) ^ and ¢ = E ( s i , t i _ l ) ^. i = 0 i = 0

It is easily seen that ¢ - ¢ _]_ C ( S ) + C(T ) . Also, ¢ and ¢ are nonnegative functionals.

Hence,

¢ ( x + y) < ¢ ( g ) and

This is the inequality which was to be proved.

¢(k) < ¢(x + y).

6.9 L E M M A . I f the parameter a in the algori thm satisfies a >_ dis t (z ,W), then

y . > c - -211KII - Ilkll ~or all ~.

P R O O F . It is clear that Yo > C. If the set

Jn = {t : y , ( t ) < y , - l ( t ) }

is nonempty, and if v E Jn then by 6.3, there is a path for which r = t~ and

y. (~) : yo(to) + ~ [ g ( ~ , , t,) - k(~,, t , _ l ) ] . i = 1

By adding two appropriate points, (so, to) and (So, t - l ) , we create a closed path. Then

by 6.8,

n

Yn (r) = Yo (to) + E [g (s,, t,) - k (s,, t i -1 )] - g ( s0 , to ) + k (so, t -1 ) i = 0

> yo(to) - K(so , t o ) + k(so , t_~)

-> - I J y 0 l l - [IKI1- Ilkll >- - 2 I J K I I - I l k l l - C,

From this it can be proved that y~ >_ C. In order to do so, suppose on the contrary

that Yn(~) < C for some ~. By the previous analysis, it follows that ~ ~ Jn. Hence

72

Y,~-I(~) = Yn(~) < C. By repeating this argument n times we arrive at the conclusion

tha t yo(~) < C, which is absurd.

If one of the sets ,In is empty, then the algori thm stops. The preceding J ,~-I is

nonempty, and the previous argument shows that y,~_ 1 > C. Since yo _> yl -> ... -> y,~-1 =

y,~ the proof is complete. •

6 .10 T H E O R E M . / f the parameter c~ in the algorithm is set equal to dis t (z ,W),

then ei ther the algorithm terminates and yields a solution as in 6.5, or it produces se-

quences {x,~} and {y,~} which converge uni formly and monoton ica l ly to funct ions x E C ( S )

and y e C ( T ) for which tlz - f o (ghx + ghy)ll = dist(z , W ) .

P R O O F . Theorem 6.5 takes care of the terminat ing case. In the other case we have

x0 <_xl_<x2 < . . . and Y0 >_Yx >-Y2-> ....

By 6.9, y , > C for all n.

The sequence {yn }, being bounded from below and nonincreasing, converges pointwise

to a function y E eoo(T). Since the sequence is equicontinuous (by 6.7), we have y E C ( T ) .

In order to see tha t the sequence {x,~} is bounded from above, we recall an inequality

from the proof of 6.5:

- ~ < z - f o (xngh + y,~gh).

From this we obtain

x,~ + y,~ <_ i f - 1 o (z + c~)]/gh = K.

Since {Yn} is bounded from below, {x,~} is bounded from above. We now conclude that

the sequence {x,~} converges uniformly and monotonically (upward) to an element x of

c(s).

In order to see tha t xg and yh provide a solution to the approximat ion problem, take

the limit in an inequality above to get

- a < z - f o (xgh + ygh).

Likewise, from another inequality in the proof of 6.5, we have in the limit

z - y o ( z g h + y g h ) < ~ .

Since a = dist (z, W), this yields Nz - f o (xgh + ygh)l[ < dist (z, W). •

73

The preceding theorem establishes the existence of a solution to the approximat ion

problem posed at the beginning, viz., to find x* E C(S) and y* E C(T) to minimize the

deviation

suplzCs , t) - / (x*Cs)hCt ) + Y* Ct)gCs)) l" 8,t

The solution pair is defined by x* = xg, y* = yh, where x and y are the limit functions

f rom the yon Golitschek Algorithm.

It should be noted tha t the constructive proof given by the algori thm establishes the

existence of a solution pair x*, y* with the proper ty that x*/g has the same modulus of

continuity as k, and y*/h has the same modulus of continuity as K. (See 6.7.)

In practice the minimum deviation p is not known in advance. A binary search

procedure can be used to determine p, or a close est imate of it. For this purpose, the

algori thm should be modified by adding another stopping criterion; viz., in step i, stop if

inf, y~(t) < C - - 2 [ [ K [ [ - Ilk N. If this stopping condition is met, it signifies tha t a < p, in

accordance with 6.9. On the other hand, if a < p then this stopping criterion will be met

at some stage, in accordance with the following lemma.

6 .11 L E M M A . I[ the paramete r a in the algorithm satist~es a < dist(z, W), then

at some step, inft yn(t) < C.

P R O O F . If y,~ > C for all n, then as in the proof of 6.10 the sequences x,~ and y~

converge to functions x E C(S) and y E C(T) such tha t

I [ z - f o (ghx + ghy)] I <_ a.

Hence a :> p. ]l

The binary search algorithm starts with the interval [0, [[z - f o 0[[]. In the general

step, an interval [a, b] will be available from the previous step. It will satisfy 0 < a < p _<

b < ] [ z - f o 0][. Then we set a = l ( a 4-b) and apply the algorithm with this value of a. If

a < p, the algorithm (with both stopping criteria) will stop with inft y~,(t) < C. If a > p,

the algori thm will stop with y , -- Y,~-I- If a = p, the algori thm may not stop, but will

produce a solution in the limit. In the first case of stopping, we replace [a, b] by [a, hi.

In the second case of stopping we replace [a, b] by [a, a]. In bo th of these cases we begin

afresh with the new interval.

74

C H A P T E R 7

T H E L 1 - V E R S I O N O F T H E D I L I B E R T O - S T R A U S A L G O R I T H M

In this chapter, S and T denote compact Hausdorff spaces. As usual, C(S) and C(T)

are the spaces of real-valued continuous functions on S and T respectively, with supremum

norms written as [I II~. We assume, in addition, that regular Borel measures # and v have

been prescribed on S and T respectively. The product measure on S × T is denoted by a.

It is assumed that /z(S} = ~,(T) = 1. Our interest now is in L~-approximation problems

in C(S × T), where the various Ll-norms are

I1~111 = f t~(s)t dr

tiulll = f ly(t ) i ~- II~II1 = S tz (s , t ) l d(r

e c ( s )

y E C(T)

z C C(S x T).

Let G and H be finite-dimensional subspaces in C(S) and C(T) respectively. Then

there exist Ll-proximity maps

A°: C(S}-~G, B ° : C(T)---~H.

Thus for a l l x E C ( S } , g c G , y e C ( T ) , a n d h E H ,

tl~ - A % t l l <_ il~ - gftl a n d ttY - B ° Y l i l -< IfY - htf~.

These maps are extended in the standard way to C(S × T). Denoting the extensions by

A and B, we have then

(az)(s, t} = (A°zt)(s) (Bz)(s, t) = (B°zs)(t).

Here zs and z t are the sections defined by

z ' ( s ) = zsCt) = z(s , t).

Note that in general Az will not belong to C(S x T) unless we assume the continuity of

the map A °.

7.1 I, EMMA. Let A ° : C(S)-+*G be an L1-proximi~y map onto a finite-dimensional

subspace G. I f A 0 is continuous in the supremum norm then A (as defined above) is an

Ll -proximi ty map of C (S × T) onto G ® C(T) , and is continuous in the supremum norm.

P R O O F . T h a t A is a cont inuous m a p of C ( S × T) into itself is established by 11.9. The

images Az lie in G ® C(T) since (Az) t = A°z t E G. If u E G ® C(T) then u t E G for all

t. Hence

tlz ~ - A ° z ~ l l l -< llz ~ - ~' lL1.

Expressed otherwise,

f , z ( s , t ) - ( A z ) ( s , t ) [ d s < f , z ( s , t ) - u ( s , t ) l d s .

W h e n bo th members of this inequali ty are in tegrated with respect to t, the result is

I l z - Aztl I <_ l l z - ull 1. •

Assuming now tha t A ° and B ° axe continuous Ll -p rox imi ty maps , we conclude tha t

the same is t rue of their extensions, A and B. The Ll-vers ion of the Di l iber to-Straus

Algor i thm then reads:

;go=z, Zk+l = Zk--Wk, wk = AZk (k odd), wk = B z k (k even).

We shall Mso require tha t A ° (x + g) : A°x + g and B°(y-4-h) = BOy + h whenever g E G

and h E H.

7 .2 L E M M A . I f u E G ® LI (T ) , v E L I ( S ) ® H, and u + v E C ( S x T), then

u e C ® C(T) and v e C(S) ® g .

P R O O F . It suffices to prove tha t v is continuous. By 11.11 we can select b io r thonorma l

bases, {gi, ¢i}~ for (G, II [11) and {hi, ¢,~.mj1 for (H, l[ ]]1). Write v = ~{m__l xihi and u =

~ ,= lg iY{ , with x, e L I (S ) and y{ e L I (T) . Put w = u + v. Then

Ily, lll = f l y , Ct)l dt= f I<¢,,~qldt<_ f I1~111 dt = Ilulll. H e n c e

II~s - ~ I 1 ~ = ~ [ g , ( s ) - -< Ig, O ) - g , ( ~ ) l l ly, lI1 i : 1 1 i = 1

--- ]Mix ~ lg, O) - g,P)l. i = 1

Now

1 ~ , ( 8 ) - ~ , ( ~ ) 1 = I < ¢ , , ~,s - ~ / I ~ I 1 ~ - ~,~-I1.,_ = I I~,s - ~ s - ~ + ~o.111

_< I I ~ s - , ~ 1 1 ~ ÷ tl~.s - ,~.!11

76

If a --* s then t]w8 - walloo --4 0 by 11.7. Hence the preceding inequal i ty es tabl ishes the

cont inui ty of xi. Therefore v is cont inuous. •

7 .3 L E M M A . There exists a constant c such that each element w of G ® C(T) 4-

C ( S ) ® H has a representation w = u + v, with

u e G ® C ( T ) , v e C ( S ) ® g , andHull14-11vltl <cllwH1.

] P R O O F . By 1.16 and 11.2, G ® L I ( T ) + L I ( S ) ® H is closed (with respec t to the L1-

no rm) in L I ( S x T ) . By 11.3 there is a cons tan t c such t h a t each e lement w in G ® L I ( T ) -4-

L1 (S) ® H has a r ep resen ta t ion w = u + v wi th

u e G ® L I ( T ) , v e L I ( S ) ® H , and ltul[l+HvH1 <cHwH1.

All t h a t r ema ins to be proved is t h a t if w E G ® C(T) + C(S) ® H, then u and v will be

cont inuous. This is es tabl ished by the preceding l emma. •

F rom now on, we wri te U = G ® C ( T ) , V = C ( S ) ® H , and W = U + V .

7.4 L E M M A . Let {zk } ~ be a sequence generated by the L~-version of the Diliberto-

Straus Algorithm. For a fixed k, suppose that z - zk = u + v, with u E U, v E V, and

[lulll + Ilvllx < ~llzll~. I f k is even then

I f k is odd, then v obeys this inequality.

P R O O F . We give the p roof for even k. The o ther case is deduced by considera t ions of

s y m m e t r y . Let {gl, ¢i} and {hi, ¢ i} be b i o r t h o n o r m a l sys tems as in the p roof of 7.2. Let

u = ~ . giYi and v -= ~ xihi. Then

IIZilll = f IZi(S) l d s = f I(¢i, vs)l d a < / I I v s I ] l d~ = IivII1 < Ailzili _< AiizlI~.

Hence

Since k is even,

0 = A(zk_~ - Azk_~) = ACz~) = A C z - u - v) = A ( z - v) - u.

Therefore u = A(z - v) and u t = A ° ( z t - vt). Thus

ly,(t)l = I(¢i, u~>l -< llu~ll~ -< lll~ ~ - ,~11~ < 211z~l11 + 2llv~lll

< 211zll~ + 2~llzl l~ ~ Ilhill~.

77

Finally,

I1,~11oo : ~-~g,y~ oo s ~ - ~ l l g , lloo Ily, l l ~

7.5 L E M M A . Le t { z k } ~ be a sequence generated by the L l - ve r s ion o f the Di l lber to-

S t raus A lgor i t hm . Le t c be the cons tan t referred to in 7.3. Then for all k,

P R O O F . Define wk, uk, and vk as in Chap te r 5 (just after i tem 5.4). Then uk E U, vk E

V, z - z z k+l = u~ + vk, and z - zzk = uk + vk--1. Let k be fixed. By 7.3, there exists an

element d E U fq V such tha t

lluk - dtlx + ]lvk-~ + dllx <- cllz - z=kll~ -< 2cllzll~.

We conclude f rom this and 7.4 tha t

Now

and

z - z 2 k + ~ = =~ + , ~ = (=~ - d) + ( ~ + 6)

II~k - dill + II~k + dill = II~k - dill + II~k + ~k -- ( ~ -- d)llx

-< 2cllzll~ + II~k + ukll~ + ll~k - dll~

-< 2~llzll~ + IIz - z2k+~llx + 2~llzllx

_< 4~11z111 + Ilzll~ + IIz2k+llll

_< (4~ + 2)llzll~ _< 6qlz[[1, Anothe r appl icat ion of 7.4 yields

It follows tha t

]lvk + dll~ ~ 6cMllzll~.

tlz - z=k+ l t [~ ~ 8cMIlzll~ and llz2k+lLt~ S 9 ~ M l l z t l ~ .

The proof for zzk is similar. •

At this point , we review the assumpt ions made about the subspaces G and H. These

are f ini te-dimensional and possess L l -p rox imi ty maps A ° and B ° respectively. We assume

t h a t A ° and B ° are cont inuous in the s u p r e m u m norm and satisfy

A ° ( x + g) = A ° x + g and B ° ( y + h) = B ° y + h

whenever x E C ( S ) , y E C ( T ) , g E G, and h E H.

78

7.6 T H E O R E M . If, in addit ion to the assumpt ions o f the preceding paragraph, it is

hypo thes i zed that B ° is Lipschl tz (in the s u p r e m u m norm), then each sequence produced

by the Di l iberto-Straus A lgor i thm with the ex t ended maps A and B is contained in a

compac t set.

P R O O F . Fix z in C ( S x T) and let {zk} be the resulting sequence. Define A', B' , and

K by

A' f = A ( z - f )

B ' / = B(z - f)

g f = f ÷ B ' . f + A ' ( f + B ' f ) .

An easy calculation shows that K ( z - z2t:) = z - z2k+2. Furthermore, if u E U and v E V,

then K ( u + v) = B ' u + A ' B ' u .

Since U + V is closed, there exists (by 11.3) a constant co such that each element

Z - - Z 2 k has a representation

Z - - Z2t: ~ . a t : + bt:

in which

ak e u, bt: e V, and Ilat:ll~ + Ilbt:lloo ~< c011z - z2kll~,

By the preceding lemma, the sequence llzt:lloo is bounded, and so is the sequence Hakllo~-

Since B ° is Lipschitz, B ' is compact as an operator from U to V by 2.23. Hence { B ' a k }

lies in a compact subset of V. Since

z - z2t:+2 = K ( z - z2t:) = K(at: + bt:)

= B ' a k + A 'B 'a t : ,

and since A' is continuous, we see that {z - z2k} lies in a compact subset of U + V. From

the equation

z2t:+l = z2t: -- Bz2t:

we conclude that the sequence {z2k+l} also lies in a compact set. Hence {zk} lles in a

compact set. U

The next two results hold for measure spaces without topology.

7.7 L E M M A . Let (S, A, p) be a ~ni te measure space. Let f be a measurable func-

tion, pos i t ive almost everywhere. Let A1, A2, .. • be measurable sets such that f A~ f ~ O.

Then I . t ( A t : ) ---* O.

P R O O F . H the conclusion is false, we can assume (by passing to subsequences) that

#(Ak) > g > 0 and that fAk f -< 2-t:. Define Bk = U~t:A~ and B = N~°=xBk. Since

79

#(Bk) >_ ]~(Ak) >_ 6, since the Bk are nested, and since t t (B i ) < co, we conclude [148, p.

17] t h a t # ( B ) > 6. Since

o o o o

k i = k i i = k

we conclude t ha t fB f = O. Hence f vanishes a lmos t everywhere on B, con t rad ic t ing the

hypotheses . []

7 .8 T H E O R E M . Let S be a measu re space, and G a l inear subspace of L1 (S). For

x E L i (S) these properties are equivalent:

(i) IlxIli = d is t i (x, S) (i.e., 0 is a best approximation to x);

(ii) f g(s)sgn ~(~) d8 < f Ig(8)l:~(~) ds for all g ~ c .

Here ~ denotes the characteristic function of {8 : x(s) = 0}.

P R O O F . This is a t heo rem of R.C. James . See [107] or [160, p. 46] for the proof . []

7 .9 L E M M A . Assume that z differs Mmost everywhere from each member of W. Let

rk be the characteristic function of {(s, t ) : sgn zk (s, t) # sgn Zk+l (s, t)}. Then f f rk [Zk [ ---+

O.

Recall t ha t tlz~ll~ converges downward. Hence tlz/~[li - t l z k + i l l i ---* O. We shall P R O O F .

prove t h a t

(1) 2 / / r ~ l z ~ l = llz~ll, - IIz~+l]ll.

Suppose t ha t k is even. Then zk+i -- Zk - B z k . By the charac te r iza t ion theo rem (7.8),

f v s g n z k + i d c r = O ( v E V ) .

Taking v = xh, for a rb i t r a ry x E C(S) and h 6 H, we have

/ ~ ( 8 ) h ( t ) sgn z~+~(8, t) d8 dt = 0.

Since x is a rb i t r a ry in C(S) we conclude f rom 11.10 and the Fubini T h e o r e m t h a t

/h(t)sgnzk+l(s,t)dt=O (hell, seS).

Since zk+i - zk E V, Zk+i(s, ") -- Zk(S, ") 6 H. Hence

f [zk+l(s,t) - zk(s,t)] s g n z k + l ( s , t ) dt = 0

8O

(s ~ S).

In teg ra t ing over S yields

f ( z k + l - zk) s g n z k + l = O.

This gives us # #

llzk+lIll = ] ] zk sgn zk+l H

=ffz sgnz.-ffz.(sgnzk-sgnzk+l, = tlzklll - - / f rtczk(sgnzk -- s g n z k + l )

n

H

= Ilzk I[1 - f f rkz (sgn, + sgn

H

The last s tep uses the fact t ha t zk and Zk+l are a lmos t everywhere different f rom 0. The

above equa t ion is a r e a r r a n g e m e n t of Eq. (1). The case when k is odd is the same, mutatis

mutandis. •

7 .10 L E M M A . Assume that z differs almost everywhere f rom each m e m b e r of W.

/_f {z~; i E I} is a uniformly convergent subsequence of {zk}, then in the notation of 7.9

lim f f r, da = O. ~el j j

P R O O F . Suppose t h a t ]If - zi[ l~ --+ 0. From the inequal i ty

and f rom the preceding l emma , we draw the conclusion t ha t

t wff : 0

Since W is closed and z - z~ E W, we have z - f E W. Thus f = z - w for some w E W,

and consequent ly the zero set of f is a null set. L e m m a 7.7 therefore appl ies to draw the

conclusion f f r~ ~ O. •

7 .11 L E M M A . Let {xn} be a sequence in C(S) which converges uniformly to a

function x. If U{s : x ( s ) = O} = O, then sgn x , converges in the metric of L1 to sgn x.

P R O O F . Firs t we shall prove t ha t ~(F=) --+ 0, where

F , = { s : sgnx,~(s) # s g n x ( s ) } .

After tha t , the p roof is comple ted by wri t ing

f t sgnxn - - s g n x I _< 2 u ( r , ) -~ 0.

81

If #(F, ,) 74 0, then for a suitable g > 0 and a suitable infinite set J of integers we have,

f o r j E J, #(Fy) >_ g. Define A,, = {s : Ix(s)l < 1/n}. We assert ~hat # (An) > 6, for all

n. To verify this, fix n and select 3" E J so tha t ]]x i - xl]oo < 1/n. Then F i c An. Indeed,

if s ~ An then either x(s) > 1/n or x(s) < - 1 / n . In the first case, xf(s) > x(s) - 1/n >

1/n - 1/n = 0. In the second case, xi(s ) < x(s) + 1/n < - 1 / n + 1/n = 0. In ei ther case,

s g n x j ( s ) = sgnx(s ) , and s ¢ Fj . Thus we have p(A,~) > p(Fy) >_ 6". Now A1 D A2 D *-'.

and #(A1) < oo. Hence [148, p. 17] we m a y conclude tha t

g - < l i m # ( A " ) = / ~ ( 5 , , = l A n ) = / z { s : x ( s ) = 0 } = 0 .

This cont radic t ion completes the proof. •

7 .12 T H E O R E M . Assume the hypotheses of Theorem 7.6. Let z be an element

of C(S x T) which differs almost everywhere f rom each element of W. Then the L1-

version of the Diliberto-Straus Algorithm produces from z a sequence {zk} such that

Ilzkllx & dist t (z ,W). Fhrthermore, the sequence {z - z,~} possesses cluster points, and

each o f them is a best Ll-approximation to z in the subspace W.

P R O O F . By 7.6, {zk} lies in a compac t subset of C(S × T). Let f be any cluster point

of this sequence, and let Ilzk~ - f [ l ~ ~ 0 as i ~ oo. We shall prove tha t z - f E W and

t h a t z - f is a best approx imat ion to z in the L l -met r i c . Since z - zk E W and W is

closed, it is clear t h a t z - f E W. In order to establish the best app rox ima t ion proper ty ,

we use the charac ter iza t ion theorem (7.8). Thus it suffices to prove tha t

f / w sgn f = (w E W). 0

For this, it suffices to prove tha t

f f usgnf=Oand f f vsgnf=O (veV, ueU). These two proofs are similar, and we consider only the first equat ion. Since u(s, t) is a

sum of terms y(t)g(s) with y • C(T) and g • G it suffices to prove

f g(s) sgn f ( s , t ) = (g • G, t • T). ds 0

Fix g • G. If k is even, then zk = zk- I -- Azk-1, and so by 7.8,

An appl icat ion of 11.10 yields

f/ u s g n z k = O.

f g s g n z k ds = 0

82

for a lmos t all t E T. If k is odd, then

fgsgnz~ds=fgsgnzk+,ds+fg(sgnzk-sgnzk+,)ds

= / grk(sgn zk - sgn Zk+l) ds

where rk is as in 7.9. I t follows t h a t

f s g n z k + l ) ds dt

= f [f 2grksgnzkds dt

2 / / I g l r k ds dt <_ ~t d

-< 21tgll~ f f ~

This inequal i ty is therefore t rue whe the r k is even or odd, and so

f l f gsgnfds dt< f [ f g(sgnf-sgnzk) ds dt+ f f gsgnzkds dt

-< Ilgll~ j] Isgn f - sgnzk [ + 2[[gl[~ ]J rk.

By the preceding l emma , f f [sgn f - sgnzk, 1--+ 0. By 7.10, f f rk, --* 0. Hence

flfasgn,dsldt=O and f g s g n , a s = 0

for a lmos t all t. II

7 .13 E X A M P L E . We exhibi t a funct ion z in C ( S x T ) for which the L l -ve r s ion

of the Di l iber to-St raus A lgo r i t hm fails, in the sense t h a t l im Ilzklll > d is t l (z, W) . Here

W = C ( S ) + C ( T ) . We let S = T = [ - 1 , 11 and define

st if s > 0 and t > 0; z ( s , t ) = - s t i f s < 0 a n d t < 0 ;

0 otherwise.

W i t h the help of 7.8 it is easily seen t ha t B z = A z = 0, and so zk = z for all k. However,

0 is not a bes t a p p r o x i m a t i o n to z, since, again by 7.8, and easy calculat ions,

if we take w ( s , t ) = s + t.

83

7.14 L E M M A . Let H be a one-dimensional subspace of C(T) generated by a posi-

tive function, h. For x E C(T) these are equivalent properties:

(i) Ilx]ll = distl (x, H)

1 f h and fg(z) h < h (ii) f ro . )h <_ ~ where P(x) = { t : x(t) > 0} and N(x) = {t : z(t) < 0}.

P R O O F . By James' characterization theorem, (7.8), (i) is equivalent to

(2) 1 / A h s g n x < fz(~) IAhl (A c ~).

Here Z(x) is the set where x(t) = 0. Then because h > 0, (2) is equivalent to

f hsgnx < fz h. (3)

Further equivalences are:

(4) L h s g n x + L h s g nx < L h

and

(7) 2 /p h < f h and 2 fN h < f h. m

7.15 L E M M A . Let H be a one-dimensional subspace of C(T) generated by a posi-

tive function h. Then there exists an Ll-proximity map B : C(T)--~H such that

(i) B is monotone: Bx > By if z > y.

(ii) B(y + ;gh) = By + flh for y E C(T) and/5 E/R.

(iii) B is Lipschitz in the supremum norm.

P R O O F . Define a nonlinear functional A : C(T) --~ ~ by

A(y) = sup{a e JR: Ily - ahlI1 = distl (y, H)} .

Then define B by By = A(y)h. Since the set of best Ll-approximations to y in H is a

compact convex set, A(y)h is one of the best approximations of y.

84

In order to prove (i), let x,y E C(T) with x > y. It suffices to show that A(x) > A(y).

If A(x) < A(y), then A(y)h is not a best approximation to x. By 7.14, either

Since x > y,

Since A(z) < A(y),

f ~ > ~ f ~ or f h > ½ f h . P(~-x(~)h) N(~-x(y)h)

N(x - A(y)h) c N(y- A(y)h).

P ( x - ~(y)h) c e ( ~ - ~(x)h) .

Hence either

f f h > h or h >

P ( x - A(x}h } N ( y - A{y)h )

By 7.14 again, either A(x)h is not a best approximation of x or A(y)h is not a best

approximation of y.

For part (ii}, we note that if IIx - ahlli = disti {x, H), then

tlx + •h - (o~ + fl}htli = disti {x, H) = d ish (x +/3h, g}.

In other words,

fl + { a : fix-- ahll i = disti (x ,H)} c { a : llx + / ~ h - ahlli = distl (x + fib, H)}.

It follows that ~ + a(x} <_ ~(z + ~h) . If we replace Z by - ~ and then z by x q-/~h we get

~(x + Zh) _< a(~) + Z. Hence

~ (z + Zh} = ~(z) + # and B(z + Zh) = B z + Zh.

For part (iii}, we assume without loss of generality that h > 1. For any x and y in

C(T) we have

I x - Yl < I1 x - Yllo~h.

Equivalently,

-1I:~ - Yll~h ÷ y <~ x < llx - Yll~h + y.

Using parts (i) and (ii) above, we have

- I tx - Yll~h + By <_ Bx <_ Ilx - y l l~h + By.

85

Equivalently,

and

- l l x - yllooh ~ B~ - By < 11~ - Yllooh

liB:= - Bylloo ~ IIx - Ylloo Ilhlloo. •

7.16 T H E O R E M . Let S and T be compact intervals on the real line, each with

Lebesgue measure. Let G be a finite-dimensional Haax subspace of C(S), and let H be a

one-dimensional Haar subspace of C(T). Let z be an element of C(S x T) which differs

a~most e y e . w h e r e from each element o f W = C(S) ~ H + C ~ C ( T ) . Then the Ll-version

of the Diliberto-Straus algorithm produces a sequence { zk } which possesses cluster points,

and IIzklll £ distl (z, W).

P R O O F . By a theorem of Jackson [160, p. 236], each element of C(S) has a unique best

L l -approx imat ion in G, and each element of C(T) has a unique best L l - approx ima t ion in

H. Hence G and H have uniquely-determined Ll -proximi ty maps , A ° and B °. It follows

that

A°(x + g) = A % + g (x e C(S), g e e )

and tha t B ° has the analogous property. The Ll-cont inui ty of A ° is guaranteed by a

theorem from [104, p. 164]. Now if {x,~} is a sequence in C(S) converging uniformly to x,

then {xr,} converges to x in the L l -norm, and so {A°x,~} converges to A°x in the L l -norm.

Since the supremum norm and the L l -no rm are equivalent on the finite-dimensional space

G, {A°x,~} converges uniformly to A°x. Hence A ° is continuous in the supremum norm.

Since H is a one-dimensional Haar subspace and T is an interval, H is generated by a

positive function. By 7.15, B ° is a Lipschitz map. The conclusions of the theorem now

follow from 7.12. •

7 .17 L E M M A . Let H be a 6nite-dimensional subspace of C(T) such that each

element of C(T) has a unique best L~.approximation in H. Let B ° : C ( T ) - ~ H be the

Ll-proximity map, and B its extension to C(S x T). Let fk E C(S × T) and fk --* f

uniformly. If f is different from 0 almost everywhere and if

then B f = O.

P R O O F . By 7.11,

f f h(t) sgnfk(s,t)dt ds--*O ( h e l l )

f f Isgn f - sgn fkl d~ ~ O.

86

Therefore

/ ] / hsgn f dt ds < / ] f h(sgn f - sgn /k) dt ds + f ] f hsgn fk dt ds ~ O.

This shows tha t f h(t) sgn f ( s , t) dt = 0 for almost all s and for all h E H. For all x e C ( S )

we therefore have

f f x(s)h(t) sgn /(s, t ) = d* ds 0.

By the Characterizat ion Theorem (7.8), 0 is a best L , -approx imat ion of ] f rom the sub-

space V = C ( S ) ® H. Since best L , -approx imat ions in H are unique, the same is true of

V, by 2.11. Hence 0 is the best L , -approx imat ion of f in V, and consequently, B f = O. •

' / .18 L E M M A . In the Ll-version o[ the Diliberto-Straus Algori thm assume that A °

and B ° a;re nonexpansive in the supremum norm. Then

llz, -zo l loo >_ IIz2- z , lloo ~> Iiz3 -z~ l loo ~ •

P R O O F . Observe first tha t the extension A is also nonexpansive since

[[A f , - A f2[[oo = sup sup [(AI,)(8, t) - (A f2)(s, t)l t 8

= sup sup [(A°f~)(s) - (A°f~)(s)[ 8

= sup []A°f~ - A°f~[Ioo t

_< s u p [[f t -- ftIlo¢ t

= Ilfl -/211o0.

Now let k be an odd integer. Then

]]zk+1 - zkll = IIAzkll = IIAzk -- Azk-,ll < llzk -- zk-*]l.

The proof for even k is similar. |

7 .19 L E M M A . Assume the hypotheses of 7.12, and assume that A ° and B ° are

nonexpansive in the supremum norm. Assume also that best Lrapprox ima t ions in G and

H are unique. I f z is an element of C ( S x T) which differs almost everywhere from each

member of W, then in the aIgorithm, [[zk+l - zkI[oo ---* 0.

P R O O F . By 7.6, the sequence {zk} is bounded and equicontinuous. Let f be a cluster

point of the sequence, and let zk~ --* f . By 7.12, z - f is a best L l - approx ima t ion of z in W,

and so 0 is a best L l -approx imat ion of f in W. Consequently 0 is a best L , - approx ima t ion

of f in U and in V. Since best L l -approximat ions in U and V are unique, it follows from

87

2.11 t h a t A / - - B f -- 0. By the fact t ha t A is nonexpans ive (see the p reced ing proof ) we

h ave

S imi l a r l y Bzk~ ---40. Now by the preceding l emma ,

[[Bzo[Ioo > HAz~[[~ >_ [[Bz2Hoo > Uaz3[[oo >_ '"

and so i t follows t h a t Bzk ---* 0 and Azk ---* O. Since zk+1 - zk is e i ther Azk or Bzk , we

conc lude t h a t IIz +, - zkll , o •

7 . 2 0 T H E O R E M . Let G and H be the one-dimensionalspaces of cons~an~ functions

in C(S ) a n d C(T) respectively, so that W : C(T) + C(S) . f f z C C ( S × T) and differs

almost everywhere f rom each element of W, then the sequence {z,,} gene ra t ed by the L1-

version of the Diliberto-Straus Algorithm converges uniformly, and z - l im z~ is a best

L~-approxlmation of z in W.

P R O O F . T h e L l - p r o x i m i t y m a p B ° : C(T)---~H desc r ibed in 7.15 is Lipschi tz . An ex-

a m i n a t i o n of the p roof of 7.15 shows t h a t in fact B ° is nonexpans ive . (The Lipschi tz con-

s t an t in 7.15 is [Ihll~.) S imi l a r ly there is a nonexpans ive L ~-p rox imi ty m a p A ° : C ( S ) - ~ G .

We define A and B by the usua l ex tens ions of A ° and B °. The e x t e n d e d m a p s , defined

on C ( S × T), are also nonexpans ive . (See the p r o o f of 7.17). F i x i n g z E C ( S × T), we p u t

A'w = A ( z - w) and B'w = B ( z - w), where w E W. Define Uk and vk as in C h a p t e r 5, j u s t

before 5.5. Then we have Uk+l = A 'S ' uk . As in 5.5, l u k ( s ) - uk(a)l < Ilzs --z~[Ioo , so t h a t

{uk} is equ icont inuous . Let f be any c lus ter p o i n t of {zk}. T h e n as in 7.19, A f ---- B f -- O.

Since W is closed, f = z - u - v for a p p r o p r i a t e u E U and v E V. The ca l cu l a t i on in 5.7

shows t h a t {uk} is bounded .

Now we know t h a t {uk} is b o u n d e d and equicont inuous . By the Ascol i Theorem,

th is sequence con ta ins a (un i formly) convergent subsequence, say uk~ -~ u*. Since A ' B ~ is

con t inuous and uk+l = A 'B 'uk , we have uk,+l --* A 'B'u*. By 7.19, z k + l - zk --* 0. Since

Z 2 k ~ Z - - U k - - V k _ 1 and Z2k.~l = Z - - U k - - Vk~ (8)

we have

vk - - v k - z = Z2k - - z2k+1 ---+ 0.

Similar ly , uk - uk -1 --~ 0. Therefore uk~+l --+ u*, and u* = A'Bru *. Then we have

This shows t h a t the sequence {Uk} i t se l f converges to u*. Since vk = Bluk, vk --+ Btu *.

Using (S), we conclude that z-z By 7.12, I11 = aistl (z, W). •

88

7.21 L E M M A . The constant M in 7.4 has the property

IIACz - , . , ) 1 I oo _< M ( l I z l t ~ + I1,-'111) ~ E C(S x T) , v E V.

P R O O F . Put u = A ( z - v ) . Let

u ( s , t ) = ~ g , ( s ) y , ( t ) and v ( s , t ) = ~ x j ( s ) h i ( t ). i=1 j = l

As in the proof of 7.4 we have ]Ix, l]1 <_ HvN1. Hence

] = 1 3'=1 y = l

Furthermore

I1,~11, = llA~Cz '~ - ,.,~)111 ~ 211 z'~ - ~'~111 ~ 211z~111 ÷ 211~'~111

_< 211z~ll~ + 211,/111 _< 211~11~ + 211~'~111

<_ 2ll~tlo~ + 211,dtl ~ l lhj t l~. j= l

Again, as in the proof of 7.4,

ly,(t)l <_ llu'~llx <_ 2llztt~ + 211,dl ~ Ilh,~ll~ . j= l

Hence r~

II~iloo ~ ~ I]g, lloolly, lloo i=1

{ / < 211zll~,+2lt'II~ tlhjlloo El lg i t l oo ] : 1 i : 1

< M(llzIIoo + 1].[11). I

7.22 T H E O R E M . Let G be a finite-dimensional subspace of C(S) having an L1-

proximity map which is L~-continuous. Let H be a finite-dimensional subspace of C(T)

having an Ll-proximity map which satis~es an Loo-Lipschitz condition. Then the subspace

W - G® C(T) + C(S) ® H

is Ll-proximinal in C(S × T).

P R O O F , . By 7.1, the subspace V = C(S) ® H is Ll-proximinal. The proximity maps

A °, A, B °, B are defined as at the beginning of this chapter. Fixing z E C(S x T), we

define F : U ---* V by the equation Fu = B(z - u). Then F is Loo-compact by 2.23.

89

Select elements wk E W so tha t

lim I I z - wklll = d i s t l ( z ,W) .

There is no loss of generality in supposing tha t

Ilwkll l --< l l~k -- sit1 + llztlx --< 211zllx _< 2 I I z I l ~ .

By 7.3, we can express each wk in the following form:

wk = uk + vk where II~kllx + Ilvkllx -< cll~lll _< 2011zll~.

Now define u~ = A ( z - vk). Then

lie - ~ - vkllx = [ I z - ' ~ k - - v k - A ( z - u k - ~k)llx

_ IIz - ~k - ~kll~ = IIz - ~okll l .

Furthermore, by 7.21 we have

Ilu~[Ioo <_ M(I I zH~ + [Ivklli) <_ (2c + 1)Mllzlloo.

Now define v~ = B ( z - u~) = F(u~). By the same argument as above,

tlz - ~L - -LIIx ___ tlz - ~ - ,~ l lx _< lie - w k l l l .

S i n c e r is c o m p a c t a n d {u~,} is b o u n d e d , t h e s e q u e n c e { v ~ } l ies in a c o m p a c t set . B y t h e

continuity of A, the sequence of points u~' = A ( z - v~) lies in a compact set. The sequence

{u~ + v~} has cluster points, and each of them is a best L l -approx imat ion of z in W. The

lat ter is closed, by 11.2. III

90

C H A P T E R 8

E S T I M A T E S OF P R O J E C T I O N C O N S T A N T S

Let Z be a Banach space and W a complemented subspace of Z. The r e l a t i v e p ro -

j e c t i o n c o n s t a n t of W in Z is defined to be the real number

A(W, Z) = inf{llPi[ : P is a projection from Z onto W}.

If this infimum is attained then the projections of least norm are called m i n i m a l p ro jec -

t i o n s from Z onto W. In this chapter we are concerned with estimating A (W, Z) when Z is

a tensor product of Banach spaces. We defer until the next chapter the problem of finding

minimal projections. Some well-known results on projections may be found in Chapter 11.

8.1 T H E O R E M . Let G and H be complemented subspaces in Banach spaces X

and Y respectively. For any uniform a on X ® Y, G -~ Y + X ~ H is complemented (and

therefore dosed) in X ®~ IF. Its relative projection constant does not exceed

A(G, X) + A(H, Y) + A(G, X)A(H, Y).

P R O O F . Let P : X---~G and Q : Y--~H be projections. By 11.2, (P ®, I) @ (I ®, Q)

is a projection of X ®~ Y onto the subspace

By the uniformity of a

W = G - @ Y + X ~ H .

tI(P ®. I) ~ ( I ® . Q)II ~< ]IP]I + ]iQl] + ]IPII IIQII.

By taking an infimum as P and Q range over all appropriate projections we obtain

A(W, X Go, Y) <_ A(G, x ) + A(H, Y) + A(G, X))~(H, Y). •

8.2 L E M M A . Let A be a rectangular matrix with the property that for some pair

of integers (#, v) we have

alv = ~ a, j = ~ aq = 1. i .i i,.i

Then Ei,j I~,il > a - 2~ .~ .

P R O O F . Define

One readily verifies that

+1 f f i = / ~ o r j = ~

eO'= --1 f f i ¢ l z a n d j # v .

eij ---- 2(5{u + 6i,. -- 5#,5y~.) -- 1.

Then

la,Jl _> i , j

E eljalj i,y

~ ( 2 6 , . + 26i~ - 2 6 , . 6 ~ - 1)a,~ i j

2 E a.y + 2 E a,~, -- 2a.~ -- E a, y

2 + 2 - - 2 a ~ , ~ - 1

3 - 2a~v. |

8.3 L E M M A . Let G and H be subspaces of dimensions n and m in Banach spaces

X and Y respectively. Then G and H have bases of norm-1 elements { g l , - - . , g,~} and

{ h i , . . . , hm} with the following property. For any uniform reasonable crossnorm ~ on

X ® Y and for any element w in G ® Y + X ® H there is a represen ration

~=~-~g,®y,+~x,®h, with ~ I1:=,11 ÷ ~ Ily, II ~ 3 ~ C ' - * ' ) . i = 1 i = l i = l i = 1

P R O O F . Select a biorthonormal set {gi,~i} for G. Thus

g~ e c , ~ , e x * , llg~ll = lI~,ll = 1, ~ , ( a ) = 5,j (1 ~ i , j ~ ~) .

Define a projection P from X onto G by the equation P x = E , = I ~ { x ) g i . Then IIPII < n.

Similarly, we construct a projection Q from Y onto H of the form Qy = ~,~=1 ¢ , (y )h ,

with IIQtl -< rn. As in the proof of 11.2 the operator P = P ® = I is a projection of X ® = Y

onto G ® Y, and IIPll _< n. Similar remarks apply to Q = I ®= Q. By 11.2, the operator

P + Q - Q P is a projection of X ®~ Y onto the subspace

W = G ® Y + X ® H .

For any w E W we write

,~ = ~,,., + ~ ( I - ~),.,,.

92

This is the representa t ion of w referred to in the s t a t ement of the theorem. In order

to verify the uppe r bounds given, consider the element u = Pw. It is of the form u =

S in - - 1 gi ® Yi for appropr ia te Yi E Y. We have

~C,.,) = , ~ ( ~ ) -< I1~11,~(~) -< ,",~(,.,.,,).

If ¢ E Y* and I1¢11 = 1 , then (because a is a reasonable norm) ,

~*(~o, ® ¢ ) = Iko, ll I1¢11 = ~.

Hence

But

Hence

(~, ® ¢)(~) < , * ( ~ , ® ¢ ) - (~ ) < , , ( ~ ) .

n

(~, ® ¢)(~) = ~ ~,(g~)¢(y~) = ¢(y,). j = l

¢(y,) < n , ( ~ ) .

Since ¢ was arbi t rary, we conclude tha t llYill < ha(W). A similar calculat ion with the tTt element v = Q(w - u) shows tha t v = ~ i = 1 xi ® hi wi th

Ilxill <__ m(1 + n)ct(w).

It follows tha t

II~,ll + ~ Iluill < [m=(~ + n) + n ~] ~(,~) < 3n~m~( ,~) . • i = l i = l

8.4 L E M M A . Let G be an n-dimensional subspace in a Banach space X, and let

H be an m-dimensional subspace in a Banach space Y. Let {xi, toi}i=lk and {y~, ¢i}~=le be

biorthonormal systems in X and Y respectively. Assume that

(i) Ett=l I~,,(a)l < ~11911 (9 e G); (iO E~=~ I¢~(h)l < cllhll (h E H).

Let ~ be a uniform reasonable crossnorm. / f P is a projection o f X ® ~ Y onto G ® Y + X ® H ,

then

P R O O F .

such tha t

E [((Pt, ® ¢,~) (P(xt , ® Y-))[ < c(ng + mk)3n2m2[[P[[. g t v

By s.3 there exist norm-I base~ { ~ , . . . , ~ , } for a and { h i , . . . ,hm} for H

~ v ~ v p ( ~ , ® y,) = ~ ® h, + ~ g~ ~ y~ i=l i=l

93

and

Now

]lx~vll + Hy~.~'ll <_ 3n2m 2 a(P(x~, ® yv)) <_ 3n2m 2 IIP[I. i = 1 ]----I

tar i = l i t i v

-< E I I ~°~'ll Hx~ "H ]¢v(h')l -< 3n2m21lP]l E ] ¢- (h , ) l iD~ iDtJ

~-3n2 m211PH E cllh'll = 3n2m2 mkcllPIl" Iti

A similar calculation gives

t in i = 1

The two estimates together yield the one in the Lemma. •

8.5 L E M M A . Let G and H be subspaces in Banach spaces X and Y respectively.

Let c~ be a uniform reasonable crossnorm on X ® Y. Assume that X and Y possess k biorthonormal systems {x,,~,}ki and {yy, ty}e i respectively such that E ,= i x, e G and

~ y = i YY E H. / f either

(i) " * l E o ' ~ ( ~ , ® ¢~)] < i whenever I',Jl = i or

(iO ~[E, j ,,i(~, ® yJ)] < 1 whenever I'oI = 1,

then every projection P of X ®~ Y onto G @ Y + X @ H satis/~es the inequality

IIPII >- 3 - 2(ke) -1 ~-~(~, ® ty ) (P (x , ® yy)). dy

P R O O F . Put ziy = xi ® yy. Then

Similarly, ~ y Pz O. = ~ 1 zo.. Thus,

= ~ ( ~ , ® ¢~)(x, ® y~) i

= ~ ~,(x , )¢~(y~) = ~ 8,,6v~ = 6v~. i i

Similarly,

F.(~o,, ® ¢,.)(Pz, s) = 6~. y

94

It then follows that

By 8.2, we conclude that

E ( ~ , ® ¢ , ) (Pzo" ) = 1. i j

t(~o,, ® %)(Pz~j) l > 3 - 2(~oj, ® ¢~)(Pz~). q

Now assume hypothesis (i). Then

'lP">-a(Pz~j) Z ( ~ e ~ - , ~ , ® ¢ v ) ( P z i J ) •

For appropriate signs Q,v this yields

[[PII > ~ ](~°t, ® ¢~)(Pzq)].

By summing this inequality over all i and j and using an inequality above, we obtain

kellPII > ~ l (~ . ® ¢~) (Pzo ) l > 3ke - 2 ~--~(~. ® ¢~.)(Pz~,.). i 3"t~u t~v

This is the inequality to be proved.

If we assume hypothesis (ii) instead of (i) we have

For appropriate e O. this yields

IlPll > ~ I(~. ® ¢~)(Pzq)l- iy

After summing over # and v we have

kgllPII >- E I(~, ® ¢~,)(Pzo.)l. i y ~ v

The remainder of the proof is as before. II

8.6 T H E O R E M . Let G and H be finite-dimensional subspaces in Banach spaces

X and Y respectively. Let a be a uniform reasonable crossnorm on X ® Y. Assume that k corresponding to each natural number k there exist biorthonormal systems {xi, ~i }i=1 for

X and {Yi, ~ k k ¢i}i=1 for Y such that ~i=1 x~ E G, ~i=1 Yi E H, and

~ I(~, ® ¢i)(z)l < ~(z) z e X ® , ~ Y . iy

95

Then each projec t ion o f X ®~ Y onto G ® Y + X ® H has norm at least 3.

P R O O F . Let g E G a n d p u t w : g ® y v . Then

l~,(g)l = ~ l','::" (g)¢.,'(Y,-')] -- ~ 1(~' ® "/'A (g ® Y-)I -< o~(g ® y~) = ]lgll. i , j i i,y

Similarly one proves tha t

I¢i(h)l _< tlhll (h e H). J

Hypothesis (i) of 8.5 is fulfilled because

~3 a ( z ) = l i3"

< sup ~ f(~, ®¢A(z)l < 1.

Now let P be a projection as described in the Theorem. By 8.5,

]]ri] > 3 - 2k -2 ~ ( ~ ® t A ( P ( x , ® yj)). i]

By 8.4, the sum in this inequality is O(k) as /¢ ---* oo. Hence in the limit we obtain

liPlt > 3. •

8.7 L E M M A . Let ~ 1 , . . . , ~ o ~ C X* and ¢ 1 , - - . , ¢ , ~ E Y*, where X and Y are

Banach spaces. A s s u m e tha t

(1) E~,=~_ [,,,:',(x)l -< Ilx[I (x • x ) (iO E,..%1 l¢,..(y)l ~ Ilyll (y e Y).

Then for all z E X ®~ Y we have

I(~,. ® ¢,)(z)l ~ .-,,(z).

P R O O F . It suffices to prove the inequality for an arbi t rary z in the uncompleted tensor

product X ® Y. Let z be such an element, and let one of its representat ions be

k

z = E x i ® y i . i : l

Then

t(~. ~ cv)(z)I ~ v

= Z ~,.Cx,)¢~C.~,)

~ IIx, II Ily, ll. i

96

If we now take an infimum over all representations of the element z and apply the definition

of the norm ~, the result is the inequality to be proved. •

8.8 T H E O R E M . Let S and T be finite measure spaces without atoms. Let G and H

be finite-dimensional subspaces containing the constants in L1 (S) and L1 (T) respectively.

Then every projection of LI (S × T) onto L~ (S) ® H + G ® Ll (T) has norm at least 3.

P R O O F . By 10.13, there exists for each integer k a measurable partit ion of S into k

sets of equal measure. Let { $ 1 , . . . , Sk} be such a partition. Define

• , (s) = c, ( s ) /~ ( s , )

where ci denotes the characteristic function of Si. Define ~i E L1 (S)* by

~,(~) = f ~(~)c'(~) ds.

One sees at once that {xi, k ~i}i=l is a biorthonormal system in L1 (S). Furthermore,

k

~ xi E G i = l

since G contains constants. It is also apparent that

i " i

In the same way, we construct a biorthonormal system {yl, ¢i } for L1 (T). By Lemma 8.7,

and 1.16,

~ Y~ ICy, ® ¢i)(z)l _< llzll z e L 1 C S × T ) . i y

By 8.6, the assertion of this theorem follows. •

8.9 C O R O L L A R Y . Assume the hypotheses of the preceding theorem. If G and H

are the ranges of norm-1 projections, then there exists a minimM projection of LI (S × T)

onto L1 (S) ® H + G ® LI(T), and it has norm 3.

P R O O F . Combine 8.1 and 8.8. •

8.10 D E F I N I T I O N . Let G be a 6nite-dimensional subspace of a Banach space X.

We say that the pair (G, X) has "Property B" if for each e > 0 there exists a biorthonormal

system {xi ,~ i} l k such that

(i) For s o m e ~ e X* , II~ll < 1 and E k - i : 1 I(~, - ~)(~)1 -< k,llgll on a ;

(ii) E ~ E ~ ,--1 ~, e a and II ,:1 ~,ll ~ k~

97

8.11 T H E O R E M . Let (G, X ) and (H, Y ) have Property B, and let a be a uniform

reasonable crossnorm on X ® Y. Assume that the biorthonormal systems referred to in the

definition of Property B have the proper ty

Then the subspace W = G ® Y + X ® H satisfies A(W, X ®~ Y) > 3.

P R O O F . Let P : X ®~, Y--~,W be a projection. Let 6 > 0. We shall prove tha t IJPtl >

3 - 6 .

Select norm-1 bases { g l , . . . , g,~} for G and { h t , . . . , hm} for H. Select e > 0 so small

tha t

(1)

where ~, and Xy are the operators referred to in 11.6. Let {x , ,~ ,}~ and {Yl, 1}1 be the

b ior thonormal systems tha t exist because of Proper ty B.

Define z~ i = xi ® YJ and we I ---- Pzi i for 1 < i < k and 1 _< j < £. As in the proof of

8.5, by the crossnorm proper ty of a , and by 8.10

~-~wiy = x~ ® Yi a w O" = <_ ek i----1 i = l i = l i---:1

1 = 1 1=I ~,1----1 J Y=l

By Lemma 11.6,

(2)

By 8.5,

m

~----= 1 v----1

k

(3) IIPII ~ 3 - 2(ke) -x ~ ( ~ , ~ ¢,)(w,y). i = 1 1 = 1

In order to est imate the double summat ion appearing in (3), we split wi] into the two

parts given in (2). The analyses of the two parts are similar, and only one is presented.

98

Taking the functional 9 given by Proper ty B, we have

i----i ]----I ij.

= + -

ij, ij~

j~ i j ,

j , i <i,

<-- ~_. 1Ig, lIek + llPll ~ kellg, H 119,11 1, 1,

= ek£~_, 119~,1t + ekglIPII ~_, [I9.11

= e k e ~ , 119.11(1 + IIPII)

< 2eke[IPII ~ 119.11.

The analysis of the other term leads to an upper bound of 2ek~iiPll ~ tI )¢-ll- The sum of

the two terms is bounded above by

Hence by (3) and (1),

D----I v----I

b~=l v = l

8.12 LEMMA. Let S be an in[inite compact Hausdorff space. Let G be a tlnite-

dimensional subspace containing the constants in C(S). Then the pair G, C(S) has Prop-

erty B as de~ned in 8.10.

P R O O F . Let e > 0. We shall construct a biorthonormal system {xi ,~i}~ with the

characteristics required for Property B.

Since S is compact Hausdorff and infinite, it has an w-accumulation point a [111,

p. 138]. Thus each neighborhood of c~ contains infinitely many points of S. Define

U i = { s e S : l g ( a ) - - g ( s ) l < 2 - ' for all g in the unit cell of G}.

99

Since the unit cell of G is equicontinuous, Ui is a neighborhood of o'. Let k > 1/e. Select

distinct points s l , . . . , s k with si E Ui. Let V1, . . . ,Vk be an open cover of S such that

sj ~ V/ when i ~ j. Let x l , . . . , Xk form a part i t ion of unity subordinate to {Vi) [148, k p. 41]. Then O< xi <_ 1, xi(sj) = 5ij, ~ i = l x~ = 1, and tlx, ll--- 1. Since

k k

( g e v , 11911<1) i = l i = 1

all the required properties are possessed by {xi,~'i}~. (Here ~'i denotes the evaluation

functional corresponding to the point si.) •

8 .13 T H E O R E M . Let S and T be int~nite compact Hausdorffspaces, and let G and

H be finite-dimensional subspaces containing constants in C(S) and C(T) respectively.

Then every projection of C(S × T) onto G ® C(T) + C(S) ® H has norm at Jeast 3.

P R O O l e . By the preceding lemma, the pairs G,C(S) and H,C(T) have Proper ty B.

In the proof of that lemma, the functions xi constructed in C(S) and the functions yl

constructed in the same way in C(T) have the proper ty

Hence 8.11 applies. •

8 .14 C O R O L L A R Y . Assume the hypotheses of the preceding theorem. If G and

H are norm-1 complemented, then there is a minimal projection of norm 3 from C(S × T)

onto G® C(T) ÷ C(S) ® H.

8.15 E X A M P L E . Boris Shekhtman communicated to us an example of a subspace

W = G ® C(T) + C(S) ® H

having a n o r m - 1 projection. Let S = [0, a ] U ( 1 } and T = [0, b] u {1}, where 0 < a < 1

and 0 < b < 1. In C(S), let G be the one-dimensional subspace generated by the function

g(s) = 8. In C(T), let H be generated by h(t) = t. Two projections are defined by

P: C(S) -~ G Px = x(1)g

Then let P = F®A I, Q = I@A Q, and L = P@Q. The map 15 is a projection o T C ( S x T ]

onto W, and a computa t ion of its norm is now necessary. First, if z E C(S × T) and

IIztl = 1 then IIZ ll = sup [z(1, t)g(8) + z(s , 1)h(t) - z(1 , x)h(t)g(s)l

8,t

= sup lsz(1,t) q- tz(s, 1) -- stz(1, 1)l. 3, t

100

The set S × T is shown in the figure.

1

b

0 0 a 1

Thus,

[[Lz[[ = m a x { s u p [z(X,t) + t z ( l , 1) - tz(X, 1)[, sup [sz(1, 1) + z(s, 1) - 8z(1, 1)[~ t6T SES

sup Is (l, t) + tz (s , I) - stz(1,1)1} (s , t )e [o,a] x [o,bl

= max{sup Iz(1, t)l, sup Iz(~, 1)1, sup Isz(1, t) + tz(s, 1) - stz(1, 1)1 } t 6 T SeS (8, t)6 [O,a] X [O,b]

< max{1, a + b + ab}.

If a = b = 1/3, we get IIL[[ = 1. •

8 .16 L E M M A . Lee S be a measure space without atoms. Let G be a finite-

dimensional subspace containing constants in Loo ( S ). Then the pair G, Loo ( S ) has Prop-

erty B as det~ned in 8.10.

P R O O F . Since S has no atoms, there is an infinite sequence of measurable sets $1, $2, .. •

in S such tha t 0 < #(Si) < oo and Si A Sj = [] when i # j (10.11). Define funct ionals ~

on Loo (S) by

= . ( s , ) -1 [ ds. ,Q

J S i

Then Ntoill = ~oi(1) -- 1. By the weak*-compactness of the unit cell in Zoo (S)* it follows

tha t the sequence {~oi}~ ° has a weak*-cluster point, ~o, and [1~11 -< 1 Since G is finite

dimensional , the sets

Ui = {%b E L,~(S)* : [~o(g) - %b(g)[ < 2 - i for g E G, Ilgll -< 1}

are weak*-ne ighborhoods of ~. Select an increasing sequence of integers vl , v 2 , . . , such

tha t ~o~i E Ui.

Now let e > 0. Select k > 1/e. Define x 2 , . . . , x k in Lo~(S) as the character is t ic

funct ions of S , 2 , . . . , S , , . Let A = S\(S~, 2 U . . . U S,k) , and let x~ be the character is t ic

101

k function of A. Then {x{, ~v,}~ is a biorthonormal system. Furthermore, ~{=1 xi = 1 C G.

Finally, if g E G and llgll -< 1, then

k k 1 1

i-----1 i : l

8.17 T H E O R E M . Let S and T be a-fnite measure spaces without atoms. Let G

and H be finite-dimensional subspaces containing constants in Loo ( S) and L¢¢ ( T) respec-

tively. Then each projection of L¢~ ( S)®~ L~ (T) onto the subspace G®Lo~ (T)+ L~ ( S)® H

has norm at least 3.

P R O O F . The pairs G, L ~ (S) and H, L ~ (T) have Property B by 8.16. The biorthonor-

real systems {x,,fo,}~ and {yi, ¢i}~ which one constructs as in the proof of that lemma

have the additional property

~"~eiyxi ® yj _< 1 ~3

By 1.53,

By 8.11, the result follows. |

eij] = 1.

= ~.. e i jx i®yj <_ 1. ~3

8.18 C O R O L L A R Y . Assume the hypotheses of 8.17. Then each projection of

L ~ ( S × T) o n t o

c Loo (T) + Loo (S) H

has norm at least 3.

102

C H A P T E R 9

M I N I M A L P R O J E C T I O N S

In the previous chapter we discussed lower bounds for projection constants. In this

chapter we show that some of those lower bounds are attained, and also investigate minimal

projections in a greater variety of spaces. The key to much of what takes place is a general

theorem of Rudin [150]. We begin by establishing this result.

The setting for this theorem is as follows. We have a Banach space X and a compact

topological group G. Defined on X is a set ~ of bounded linear, bijective operators in such

a way that ~ is algebraically isomorphic to G. The image of g • G under this isomorphism

will be denoted by Ag. We shall assume that the map (g, x) ~-* Agx from G x X into X

is continuous. A subspace Y of X is said to be i n v a r i a n t under G if A g Y c Y for all

g • G. An operator B : X --* X is said to c o m m u t e with G i£ BAg : A a B for all g in G.

Throughout the remainder of the exposition we will write g in place of Ag and use such

notation as Ilglt when we mean liAgt[.

9.1 T H E O R E M . Let X and G satisfy the above hypotheses, and let Y be a closed

subspace o£ X which is invariant under G. //' there exists a bounded projection P of X

onto Y, then there exists a bounded projection Q of X onto Y which commutes with G.

P R O O F . We define the sets Ek,x C G by

Ek, = {g • a : Ilgxll k Ilxll}

where x E X and k = 1, 2, 3, . . . . Since the mapping g ~ gx is continuous Ek,x is closed.

Then the intersection of all such sets over x E X, which we shall denote by Ek, is also

closed. Clearly,

Ek = {g e G : lfgll -< k},

and G = U~= 1 Ek. The Baire theorem may now be applied to conclude that there exists

an rn and an open set V c Err,. Now each element of G is in some translate of V and

so the translates gV form an open cover of G. Since G is compact, finitely many of these

translates cover G, say g l V , . . . , g,~V. Now for any g E G we have g E giV for some i; thus

for some v E V

Ilgll = IIg ' ll <- m a x Ilgsll I1' 11 < Mrn. l < Y < ~

Hence the o p e r a t o r s in A are un i fo rmly b o u n d e d . Define an o p e r a t o r Q on X by

Qx = / g- 1Pgx dg.

G

Here dg denotes the Haa r measure on G, no rma l i s ed so t ha t the measure of G is one.

Now for a fixed x, g - l p g x is a con t inuous m a p p i n g f rom G into X. Hence Q is

wel l -def ined, l inear , and ItQN <- M m IIptl.

For any x E X we have Pgx E Y for all g C G and, since Y is i nva r i an t unde r

G, g - l P g x E Y. Since Y is closed, it follows t h a t Qx E Y. Fur the rmore , if x E Y, then

gx E Y and so Pgx --~ gx and g - l P g x = x. Hence Qx -= x, and we have e s t ab l i shed t h a t

Q is a p ro j ec t ion of X onto Y.

F ina l ly , we show t h a t Q commutes wi th G. F i x go E G and pu t h -= ggo so t h a t

g-1 = goh-1. Using the fact t h a t Haa r measure is t r a n s l a t i o n - i n v a r i a n t we have

Qgox = / g - lPggoxdg= / g o h - l P h d h = goQx. "

G

We shal l now take S and T to be finite sets which we sha l l iden t i fy as

S = { 1 , 2 , . . . , n } and T = { 1 , 2 , . . . , m } .

In i t ia l ly , we cons ider the l inear space X of all r ea l -va lued func t ions on S × T. Let U be

the subspace of X cons is t ing of func t ions which d e p e n d on ly on the s -va r i ab le while V is

the subspace of func t ions d e p e n d i n g on the t -var iab le . Let W = U + V. Of i m p o r t a n c e

in the fol lowing discuss ion wil l be the fact t h a t a func t ion x E X is in W if and only if x

satisfies the " four -po in t ru le" :

(1) x ( i , j ) = x ( k , j ) + x ( i , ~ ) - x ( k , £ ) l <_i,k<_n, l <_j,£<_rn.

This is e s t ab l i shed as follows. If x E U or x E V, it is e l e m e n t a r y to ver i fy (1). Hence (1)

ho lds for x E W. On the o the r hand , if x be longs to X and sat isf ies (1), and if (k,~{) is a

fixed po in t in S × T, then (1) d i sp lays x as the sum of a func t ion of j , viz. x(k, j), and a

func t ion of i, viz. x(i, ~) - x(k, £).

It wil l be convenien t to use m a t r i x t e r m i n o l o g y for the func t ion x, so t ha t , for example ,

the values x(i , j ) , 1 <_ j <_ m, will be referred to as the i th row of x. Define p e r m u t a t i o n s

~rij which in t e rchanges rows i and j in x and vii which in te rchanges co lumns i and j .

These genera te a finite g roup which we will deno te by G. The e lements of G are

a s soc ia t ed w i th t r a n s f o r m a t i o n s on X in the obvious way so t h a t we will take AN to be

def ined by

(A,~x)(i , j)= ( x o r ) ( i , j ) ( r • G ) .

104

We will consider X to have a norm which constrains the maps A to be isometries. The

hypotheses of 9.1 are now met. The subspace W is invariant under the t rans format ions

An. The mapp ing (x, g) ~-~ Agx is cont inuous if G is given the discrete topology, since for

a n y x o E X a n d g E G t h e set

( ( ~ , ~ ) : II~0 - ~ll < ~}

is carried by this mapp i ng into the set

{ z : l[Aoxo - zll < ~}.

In fact we claim a s t ronger result than 9.1 holds.

9 .2 L E M M A . There is a unique projection Q : X --~ W which c o m m u t e s with G.

P R O O F . Take ers to be the funct ion defined by

and set

e~s (i, j ) = 6r~ 6sj

~,~ = Q ~ = ~ = ~ ( i , y ) ~ j .

i ~ 1 ]-~i

Now ~'rs e l i -- e l i whenever r, s _> 2, and recalling tha t we require

~rr8 all : Irrs Q e l l : Q~rrseii ---- Qeii ---- aii

we must have

a l l (r, j ) = aii (s,] ') ,

By a similar a rgumen t using rr~ we obta in

a l l (i, r) = ~ (i, ~),

2 < _ r , s < _ n , l < ] < _ r n .

These two equat ions give

Now using

l < i < n , 2 < _ r , ~ < m .

b i - - - - j = 1

al l ( i ,3") = c 2 < i < n, j = 1 d i = 1 , 2 < _ j < _ r n e 2 < i < n , 2<_3"<_rn

at8 = Qers ~- Qvsi rri eli = rsi rri Qeii = rsi rri all

105

we have b i = r , j = s

~ ( ~ , s ) = ~ ~#,-, j = s d i = r , j # s " e i ~ r , j ~ s

Now since ars E W, the four-point rule gives

b + e = c + d .

Also the two equations

i = l i = 1

give

and Q e~s = E eii 3"=1 j = l

b + ( . - 1)c = 1

d + ( n - 1)e = 0,

and

b + ( m - 1)d = 1

c+(m-- 1)e=O.

These five equations have the unique solution

n + m - 1 m - 1 n - 1 -1 b - , c - - - - , d = - - , e = - - ,

n m n m n m n ~

which defines the projection Q which commutes with G. II

9.3 T H E O R E M . If P is any projection from X onto W, then the projection R

defined by I *

= / g- x Pgx dg Rx

G

is a minima~ proiection of X onto W. (Recall that ll01l = 1 for a~l 0.)

P R O O F . From the proof of 9.1 we know that R is a projection which commutes with

G. From 9.2 we know that such a projection is unique. Hence using Q from 9.2, we have

Qx = f g-lpgxdg,

G

for any projection P. Since the transformations associated with G have unit norm,

IIQxlt < f IIg -1Pgxll dx < f llg-lll IIp[[ IIgll tlxlI dx

G G

< I[Pll [lxll.

106

Tak ing a s u p r e m u m over al l x of no rm 1, we o b t a i n ]]Q]] _< IIPII. I

9 .4 C O R O L L A R Y . Let S = { 1 , . . . , n} and T = { 1 , . . . , rn} with measures ]z and

giving ~(i) = 1/,~, and ~,(y) = 1/m. Then a minimal projectio. Q from L~(S × T) onto

Lp(S) q- Lp(T) is given by

( n + m - - 1 ) / ( n m ) i=r, ]=s (rn-- 1)/(nm) i # r, ] = s

(qe,s)(~,y) = (~-- X)/(~m) i = r, 3 #

(-1)/(nm) i # r, ] # s

where er~(i,Y) = 6 . 6~j (1 _< v __ oo).

P R O O F . Observe first t h a t p e r m u t a t i o n s of the rows or co lumns of a func t ion

x e Lp(S × T)

do not a l t e r i ts norm. Hence the ope ra to r s A , have uni t no rm. Thus the p ro j ec t ion Q

ca l c u l a t ed in the p roof of 9.2 is a m i n i m a l one by 9.3. ]]

Th is resul t m a y be ex t ended to the case where S and T are finite measure spaces. To

do th is we mus t first e s tab l i sh some n o t a t i o n and p r e l i m i n a r y resul ts . We shal l assume

f rom now on t h a t (S, 1~,/z) and (T, O, u) are finite, n o n - a t o m i c measu re spaces w i th # ( S ) =

~(T) = 1. The p r o d u c t measure space (S × T, ff~, a) is then cons t ruc t ed in the usua l way.

We shal l f rom now on use Z to deno te n v ( s × T) and W to deno te L , ( S ) + L v ( T ). The

spaces in the p rev ious case (9.4), when S and T are pure ly a tomic and are each as sumed

to c on t a in n poin ts , wil l now be referred to as X,~ and IV, . Our c a n d i d a t e for a m i n i m a l

p ro j ec t ion f rom X onto W wil l be

S T SxT

It is easy to see t h a t i f p = 1 or oo then liP011 _< 3. Then 8.S and S . lS give:

9 .5 T H E O R E M . The projection Po defined above is a minimal projection from

Lp(S × T) onto Lp(S) + L v ( T ) where p = 1 or oo and S and T are non-atomic measure

spaces each having measure 1.

In the case of Loo(S x T), the hypo thes i s t ha t S and T are finite in 9.5 can be

weakened to a-f ini te . Then a m i n i m a l p ro j ec t ion is given by a fo rmula like the one above,

except t h a t the in tegra ls are over any sets A, B, A x B, all of measure 1.

107

Now for a fixed value of n take {S/}~', {Ti} • to be measurable part i t ions of S and T

respectively satisfying

#(Si) = v(Ti) = 1/n, 1 <_ i, ] < n.

This is possible by virtue of 10.13. With the aid of these part i t ions we define operators

U. : X--* X. and 17. : X. -* X by

(V.z)(i,h = n2 f f xd~ ( l < i , j < n )

SixTj and

9 . 6 L E M M A .

l < p < o o .

P R O O F . Firstly,

V , z = £ z(i,])XS, xTi. i , ] = l

The linear operators Un, 17. defined above are both of unit norm for

1 ip IIU.xll p = ~ ~ ICU.=)(~,h

i j '

1 P

*,J S i x T i

_< ~ ~ I=l,-,=XlxT, do-

Now an application of HSlder's inequality gives

IIU,,=ll p -< ~ F_,. . -" px'~, ×~, do "=~X~,×T, d~,

where, p -1 + q-1 = 1. Now

/ n2aXqS, xT j d a = n 2a-2

S x T

108

Secondly, we have

ff / / P / / IIV,,~ll p = IV~ l p : ~ = ( / , j ) x ~ , × ~ j = ~ I= ( i , j ) I ' ~X~ ,×T ,

ff , = ~ I~(i,3)1" X~,x~., -- ~ ~ - I z C ~ , h l '~ -- Ilzll '~. i3" d3'

and so 1 1

IIU.~lF < U IlxllP n(2q-2)Pq- --II~IF

H e n c e IIU.II < 1 and it is easy to see, using the func t ion x which takes the value 1

everywhere, tha t IIU.I] = 1. |

9 .7 L E M M A . / f P is any projection from X onto W then U . P V . is a projection

from X. onto W . .

P R O O F . Suppose z E X . . T h e n since P has range W, we have P V . z C W. Hence

P V . z = u + v for some u e Lp(S) , v e Lp(T). Now

S~ xTs St xTk

S, ×Tj (

= n , { / u d #

Six Tk

StxTk

St xT~ St xT~ Si xTk

= ( U . P V . z ) ( i , k) + ( U . P V . z ) ( g , j ) .

Thus U . P V . z satisfies the "four-point rule" whenever z E X . and so the range of U . P V .

is con ta ined in W . . To see t ha t U . P V . projects onto W . , take z E W . where

z ( i , j ) = ki, 1 < i, 3" < n.

T h e n

Vnz = ~ Z(i,")XS~xTi = ~ ]QXS~xTi i , ] = 1 i,3"=1

Thus V . z E Lv(S) and so P V . z = V . z . Now

=- ~ ki Xsi xT. i = 1

f f

o

SixTj i,j=l ~t

Hence U . P V . z = z. A s imi lar a rgumen t shows tha t if z E W . where

l < _ i , j < n .

z(i, j) = k., 1 < i, j < . ,

t hen U . P V . z = z and so U.PVn acts as the ident i ty on W. . •

109

9.8 L E M M A . Let Q and Po be defined as in 9.4 and 9.5. Then Q = U, PoV,.

P R O O F . It will suffice to establish Qerk = U, PoV,~e~k where e~k (i, j) = 5~ ~kj. Observe

t ha t V,e~k = XS, xTk and so

S T SxT

2 1 7~ ,;2 ~ ~ s~, t c Tk

1 1 = n -£7 s 6 S ~ , t C T k

1 1 n -~ s ~ S ~ , teTk

--I

Hence 2 n - 1

( U ~ P o y . ~ r k ) ( ~ , j ) = ~ - 1

- i

n2

i = r , 3"=k

i¢,-, i = k

i • r , ] ~ k

: ( Q ~ r k ) ( ~ , j ) . []

or i = r, ] TL k

9.9 T H E O R E M . Let S and T be nonatomic measure spaces each of measure 1.

Then the projection

P 0 : Lv(S × T) -~ Lv(S ) + Lv(T )

given by

(Pox)(s,t) = / x ( a , t ) d # ( a ) + / z ( s , b ) d ~ ( b ) - / / x(a,b) d#(a)d~,(b)

S T SxT

is a minimM projection for 1 ~_ p ~_ oe.

P R O O F . We shall only consider 1 < p < oo since p -- 1, oo have been covered in 9.5.

Given e > 0 we choose x e X wi th [Ix[[ = 1 and

IIPo~ll ~ I lPol l - ~.

By 10.15 we can find pa r t i t i ons {Si}?, {Tj}? of S and T such tha t

1 ~(S,) = ~(~ . ) = - , 1 _< i, ] -<

n

110

and such that there exists a function y of the form

y= ~ aijXs, xT¢ with IIx-ylI < e. i , ] = l

Combining 9.7, 9.8 and 9.3, we have for any projection P from X onto W and for all

z6X.

Now U.y e X., and so

U . P o Y . ~ =

U . P o V , . U . y =

Because of the form of y we have V,~U.y

U.Poy= f G

Now for (s, t) 6 Sr x Tk we have

f g-lU.PV.gz dg. G

f g-~U.PV.gU.ydg. G

= y, and so

g-lU.PV,.,gU.ydg.

S ~,3 T ~,3

S xT ,,3

_ 1

S i T ] i , ]

= - •

n ~ 3" i,3"

Hence Toy = ~ ] = 1 bl] XsixT3. We have

VnU.Poy = Vn f g-~U.PV.gU.ydg G

or

Poy = V. f g-ZU.PV.gU.ydg. G

Now

llPoll < [lpoxll + c < llPoyll + IlPo(x - y)ll ÷ ~

< llPoyll + IlPoll tlx - yl[ +

-< llPoyll + ,(ilPott + 1)

= llvnf g-ZU,~PV,-,gUr, ydgll + ~(llPoll + 1) G

< [IV,~II f [[g-XN IIU,~II IIPll [Iv,l[ llgll llu,~ll [lyl[ dg + e(llPo H + 1).

G

111

By 9.6, ]]Ur~]] -- HV,,[[ = 1. From the proof of 9.4, ]]g[[ = 1. Hence

]IPo]l < llPll Iivll + c(iIPolI + 1)

-< ]]Pll (1 + ~) + 4]lP0]i + 1).

Since e was arbi trary, we conclude tha t P0 is indeed a min imal project ion. •

112

C H A P T E R 10

A P P E N D I X O N T H E B O C H N E R I N T E G R A L

In the course of the previous chapters, we have frequently discussed approximation-

theoretic questions involving measures and integrals. In particular, we have employed in-

tegrals of functions whose values lie in Banach spaces. These integrals, known as B o c h n e r

integrals, are the subject of this expository chapter. Our intention is only to provide an

introduction to their basic theory. At the end of the chapter, there are a few results

concerning classical measure theory which were used without proof in earlier chapters.

The Bochner integral has as essential constituents a measure space (S, 4, #) and a

Banach space X. Thus S is a set, ~q is a a-algebra of subsets of S, and/z is a (countably

additive nonnegative) measure on 4. As is customary, we refer to S as the measure space,

since ~q and p remain fixed in the background. The elements of ~ are the m e a s u r a b l e

sets . A measurable set of measure zero is termed a nul l set .

A function f : S ~ X is said to be s imp le if its range contains only finitely many

points :T1, X 2 , . . . , X n in X, and if f - 1 ( x i ) is measurable for i = 1 , 2 , . . . , n. Such an f can

be written

n

f = ~ xi XE,

where XE, is the characteristic function of the set Ei = f - l ( x l ) . A function f : S ---* X is

said to be s t r o n g l y m e a s u r a b l e if there exists a sequence {f,~} of simple functions with

l i m I I f - (~) -- f (~) l l = 0

almost everywhere (i.e., except on a null set.) The following few lemmas help to explain

the concept of strong measurability.

10.1 L E M M A . Let S be a measure space, X a Banach space, and f a s trongly

measurable funct ion from S to X . For any open or closed set Y in X , f - l ( y ) can be

expressed as the union o f a measurable set with a subset o f a null set.

P R O O F . Let {fn} be a sequence of simple functions such that fn(s) -* f (s ) on the set

S ~ = S \ N, where N is a suitable null set. Let Y be closed in X, and define

E,~k = {s e S ' : dist(f , , (s) , Y) <_ l /k} .

Each set Enk is measurable, and one verifies easily tha t

o o o o

f-i(r/n s ' - N U Eo . k = l m = l r ~ = m

Hence f - 1 (y ) f~ S ' is measurable.

If Y is an open set in X, then write

f - l ( y ) A S ' = S' \ I - I ( X \ Y).

The set on the right side of this equation is measurable, by the first half of the proof. In

either case, the proof is completed by writing

f - l ( y ) = [ / - l ( r ) n S'] u [ / - X ( r ) n N]. l

10.2 L E M M A . Let S be a complete measure space, X a Banach space, and f a

strongly measurable function from S into X. Then f is measurable in the classical sense.

P R O O F . We have to show tha t f - l ( 0 ) is measurable for every open set 0 c X.

From the previous l emma we can write f - l ( 0 ) = A t2 B, where B is measurable and A is

contained in a null set. Since the measure is complete, A is measurable, and hence f - l ( 0 )

is also measurable. •

10.8 L E M M A . If f : S --~ X is measurable in the classical sense and has essentially

separable range, then f is strongly measurable.

P R O O F . Let N be a null set such t h a t / ( S \ N) is separable. Let { x . } be a countable

dense set in f ( S \ N) . For each n, a simple function fr, is defined as follows. Put X,~ =

{ x l , . . . , x n } and fix s E S \ N . If d i s t ( / ( s ) , X , ) > 1 put fn(s) = 0. If d is t ( f (s ) ,X,~) _< 1

let i be the largest integer in {1 , . . . ,n} for which dis t ( f (s ) ,X,~) < 1/i. Let j be the first

integer for which Ill(s) - xjl [ < 1/i and set f, ,(s) = xj.

In order to show that f , ( s ) ---* f ( s ) for almost all s e S, let s be any point in S \ N

and let 0 < e _< 1. Select an integer k such that 1/k < e, and select m > k so tha t

Ilzm - / (~)11 < 1/k. I f n >_ m then xm e Xn and d is t ( f (s ) , X,,) < 1/k. Hence

l l f . ( ~ ) - f (~) l l -< 1 / k < , .

Thus f is the limit almost everywhere of a sequence of simple functions, and consequently

is strongly measurable. •

114

10 .4 L E M M A . h t a sequence of strongly measurable functions from a finite measure

space to a Banach space converges almost everywhere to a strongly measurable function,

then for each e > 0 there is a measurable set of measure less t han e on the complement of

which the convergence is uniform.

P R O O F . Let f and fn be s t rong ly m e a s u r a b l e f rom S to X, w i th fn(s) --~ f (s ) a lmos t

everywhere . Let e > 0. We wil l cons t ruc t a m e a s u r a b l e set E , hav ing measu re a t mos t e,

such t h a t f,~(s) --~ f ( s ) un i fo rmly on S \ E .

By r emov ing a nul l set f rom S we o b t a i n a set S ' such t h a t f,~(s) --~ f ( s ) everywhere

on S ' . The p roo f of 10.1 shows t h a t by r emov ing a n o t h e r nul l set we o b t a i n a set S " w i th

the p r o p e r t y t h a t S " N (fn - f ) - l ( O ) is m e a s u r a b l e for each open set 0 in X and for each

n .

As a resul t of these cons ide ra t ions , we conc lude t h a t the fol lowing sets are measu rab l e :

o o

E ~ = N { s • S " : tl/,(s) - f(8)il < 1 / m }

Clea r ly E ~ c E~ n c . . •. Since fn(s) ---* f (s ) on S " , we have S " = Un=lE, ~ c o m for each m.

Since S " is of f inite measure , there exis ts , for each m, an in teger k(m) such t h a t

# ( S " \ Ek'~m)) < e/2 m.

Define the set

E =

o o

U (s"\ E;im)). m = l

T h e n E is m e a s u r a b l e a n d has measure less t h a n e. If s E S" \ E then s E E~'~.~) for al l

m, and consequen t ly the i nequa l i t y

IIf,( ) - fC )ll < 1/m

is t rue for a l l i > k(m). This es tab l i shes the un i fo rm convergence

fries) ~ f (s ) on S " \ E.

10 .5 L E M M A . Let S be a finite m e a s u r e space, X a B a n a c h space, and f : S --~ X

a strongly measurable function. Then there exists a null set E such that f ( S \ E) is

separable in X .

115

PROOF. Since f is strongly measurable, there is a sequence of simple functions g.

such that [[g.(s) - f(s)[[ --* 0 for almost all s E S. Using 10.4, we can find for each

integer m a measurable set Em such that /z(E.~) < 1/m and such that the sequence {gn} converges uniformly to f on S \ Era. The range, G., of g,~ is a finite set in X, and

U~=IG. is a countable set whose closure contains f(S \ Era). The latter set is therefore separable. Define E = A~=IE,~. Then/~(E) = 0. Also f(S \ E) is separable, being equal to U ~ = I [ ( S \ E , , ) . III

10.6 T H E O R E M . Let S be a ~nite measure space, X be a Banach space, and

f : S -* X a strongly measurable function. Then there exists a sequence of countably-

valued strongly measurable functions which converges almos~ everywhere uniformly to

[.

P R O O F . By removing a null set from S if necessary (see 10.5) we may assume that

X o o f ( S ) is separable. Suppose { -}1 is a countable dense subset of f ( S ) . For s E S, define

n = nCm, s) to be the first integer such that I l l ( s ) - x . l l < ~ . Put fro(S) = Xn. Then for

all s E S, II f ( s ) - f m (s)1I < Z It remains to show that each fm is strongly measurable. m"

Fix m, and by 10.1 write

{ s e S : l l f C ~ ) - z , lI< ~ } = C , u D i

where Di is measurable and Ci is contained in a null set, C~ say. Now put C = UiC~ and

define, for each k >_ 1,

f k ( s ) = { ~ , , ( s ) w h e n e v e r s e S \ C a n d fm(s)

Then the range of fk m is {0, Xl, X2 , . . . , xk} , and, for 1 < i < k,

(I~)-*(.,) = {s e s \ c :

={seS\C :

={seS\C : I l f (~) - z, II < ± m and r a i n Ill(s) - *Jlt ~ ~} l_<j<i

= (s \ c) n (c~ u D,) \ U(c~. u Di) j < i

=(S\C) nDi\UDi. ]<i

This last set is measurable. Also, (fkm)-l(O) is measurable, since

k

(SL)-~(0) = s \ UCS~)-~(~,). i = 1

Thus each S~m is a simple function. It is clear that SL ( ' ) - ' Sin(s) for almo,t all s e S

and so fm is strongly measurable. •

116

Note that if f is a strongly measurable function having countable range {xl, x~, . . .} ,

then for almost all s we have ( 3 O

f(8) ---- E C,(S)X, i = 1

where the ci are characteristic functions of measurable sets. Indeed, by 10.1, we can write

f - l ( x i ) = [ f - l (x i ) n N] U [ f - l ( x i ) \ g ]

for a suitable null set N. Then ci can be defined as the characteristic function of f - 1 (x¢)\N.

Now iet (S ,A,#) be a finite measure space, and f a simple function from S to a

Banach space X. Write

= c,(s)x, i = l

where the ci are characteristic functions of sets E~ E .4. Then we define the integral of f

over any measurable set E by

f f (s)ds = ~ . ( E N E ~ ) x , . E i = l

This definition is independent of the representation of f.

10.7 L E M M A . For any simple function f : S -~ X,

E E

P R O O F . It suffices to give the proof when E = S. Let f be as above, assuming in

addition that the sets Ei are mutually disjoint. Then

i = 1 i = 1

n

-- c,(s)tlx, IIds = f E ci(s)x{ ds z = l i----I

= f ll/(s)ll ds.

10.8 D E F I N I T I O N . Let S be a finite measure space and X a Banach space. A

strongly measurable function f : S -+ X is said to be Bochner integrable i[ there exists a

sequence of simple functions f,~ such that

(i) limf IIf..(s) - /(s) l l ds ---- O.

In this case we define

(ii) f f(s)ds = l i m f f,~Cs)ds. E n E

117

In order to see tha t the l imit in (ii) exists, use 10.7 to write

fE /B(s)ds - f s fm(s)ds <- fs NfB(s) - f,o(s)N ds

_< J~ iisBcs)- sCs)ll as + j~ list8) - s,,(~)ll as. Thus the sequence in (ii) is a Cauchy sequence in X.

Different sequences satisfying (i) lead to the same value for the integral in (ii) by a

similar argument.

10 .9 T H E O R E M . Let S be a finite measure space and X a Banach space. A

strongly measurable function S : S ~ X is Sochner integrable if and onSy if f s II/(s) II d8

is anite.

P R O O F . Assume tha t f is Bochner integrable, and let {fn} be a sequence of simple

funct ions wi th f s f(s) ds = limb fs/n(s) ds. Then

f ll/(s)ll as_< f ll/(s) - / , (s) l l d~+ f It/,(,)11 as s s s

which is finite for all n.

Conversely, suppose f is strongly measurable, and fs I I / (s) l l as < co. By 10.6 there

exists a sequence {fB} of countably-va lued s t rongly measurable funct ions and a null set

E such tha t i l l(s) - f~(s)l I < 1In for all s e S \ E . Since

II/B(,)II < II/(~)1t + 1In

on S \ E , we have

/ I1/.(~)11 ds <_ /II/(s)ll ds + vCS)l, s s

a n d so fs II/ ,(s)ll ds < co. By the remark following 10.6, we can write the following

equat ion, for all n and for almost all s:

In(s) = ~ xB~ cB~(~), m ~ l

where Xnm E X and c,~m is the character is t ic funct ion of a measurable set E,~,~. W i t h o u t

loss of generality, we can assume

E o i f q E B i = D for i ~ j .

For each n chose PB such tha t the set

has the p rope r ty

En m n = p n + l

f ll.r.(s)lt d~ < ~(S)/~. Fn

118

P~ Then each g1% is a simple function with Define g1% = ~ m = l Xnm C1%m.

f t l f (s)-gn(s)N ds< f Ill(8) - f~(s)H d s + fi Hf~(s)- g~(s)ll ds < 2#(S) /n .

S S S

Hence f is Bochner integrable. •

There axe many standard results from the scalar theory that can be carried over to

the Bochner integral case. As an example we give the following theorem.

10.10 L E M M A . Let S be a anite measure space, X a Banach space, and f a

Bochner integrable function. Then

(i) II f/(s)dsII < f IIf(~)ll d8 (ii) limff(s)ds = 0 as ~(E) ~ 0, uniformly in E.

E

(iii) ze f / ( s )d~ = 0 for aH measurable E, then ](~) = 0 a.e. E

P R O O F . (i) This inequality is true for simple funcitons, by 10.7. Let {f,~} be a

sequence of simple functions such that

lim/IIf1%(~) -/(~)11 d s = O.

From the inequalities

If lifo(s),, d s - f Ilf(s)l I ds < f lllf~(s)ll-llY(~),l d~

_< /Ill.(8) - f(8)ll we conclude that

/II.ro( /lt - - /tl.rc. )I1 lira ds.

Hence

< l im f Ilf.(s)dsll = / Ilf(s)ll ds.

(ii) Given e > 0, let g be a simple function such that

f ll/(s)-g(~)ll < , / 2 . ds

Suppose that 9 has the representation n

g : ~ zi ci i = 1

where each ci is the characteristic function of a measurable set A~. Let E be any measurable

set such that 1%

1 p,(E) < ~ e / ~ I1~:'11" i = 1

119

O

II IA

II

II II

II II

IA

t~

I

÷ a.

IA

-t-

z ~

~ ~

~ +

~ +~

,

~ ".

-."

"4-

+ +

+

t~

I A

v O

~a

~.

[A

II

IA

[A

I

+ ~

- +

I

÷ A • t

O

We conclude this section about Bochner integrals by describing briefly some spaces of

Bochner integrable functions. If 1 < p < oo then Lp(S,X) will denote the Banach space of (equivalence classes of) strongly measurable functions f : S --~ X such that

f II/(8)IIP d8 < ~. S

The norm in Lp (S, X) is defined to be

The essentially bounded strongly measurable functions f : S -+ X form the Banach

space Lco (S, X) with norm

Ilflloo = ess sup IIf(s)]l.

We now revert to the case X = ~ for the remainder of this appendix. Our purpose is

to establish some measure theoretic results which were used in earlier chapters but were

secondary to the main issues contained there.

Recall tha t an a t o m in a measure space S is a measurable set A of positive measure

having the proper ty tha t each of its measurable subsets has measure either 0 or #(A). A

measure space is said to be n o n - a t o m i c if it contains no atoms.

10.11 L E M M A . Let S be a nontr iv ia I non-a tomic measure space. /if e > 0, then

there ex is t s a measurab le set E such tha t 0 < # ( E ) < e.

P R O O F . Since S is non-atomic, there exists a measurable set S ' such tha t

o < .(s') < .(s).

Then # (S ' ) < co. Let E t be a measurable set in S ' such that 0 < # ( E t ) < # (S ' ) . Then

# ( S ' \ E t ) > 0, and there exists a measurable set E2 c S ' \ E1 such tha t

0 < #(E2) < / z ( S ' \ E l ) .

Similarly, we find E3 C S ' \ (El u E2) with

0 < #(Ez) < # ( S ' \ (El u E2)).

Proceeding in this way, we construct a disjoint sequence of sets Er, C S ' with #(E,~) > 0.

Hence

E # ( E " ) = # E , < # ( S ' ) < c o . n=l n = l

Clearly # ( E , ) < e for sufficiently large n. •

121

10.12 L E M M A . Let S be a finite and non-atomic measure space. / f 0 < ~ < #(S) ,

then there is a measurable set E such that # (E) = 0.

P R O O F . For any measurable set A, define the monotone set function

~(A) -- sup{#(H) : H measurable, H c A, # ( H ) <: 0}.

Using 10.11, select A1 so tha t 0 < # (A, ) _< 0. Select A2 c S \ A, so tha t

½/9(S \ At) _< #(A2) _< O.

Continuing inductively, we obtain a disjoint sequence of sets A,~ such that

i = l

and

By the monotonici ty of 8,

fl S \ U A , <_~ S \ A, < 2 # ( A , + , ) . i-----1 i = l

Since the A, are disjoint, ~,°°=1 #(A,) ~ #(S) , and hence /,(A,~) --+ 0. Therefore OO f l (S \ U,=IA. ) = O. By 10.11,

c o

# ( S \ U A , ) = 0 and E / z ( A ' ) = / * ( S ) > 6 " i = l

oo A Hence there is a subsequence such that ~{=1 # ( , ~ . ) -- ~" Consequently

, ( u ~ l A . , ) = 0. []

10.13 L E M M A . Let S be a finite non-a tomic measure space. Then for each natural

n u m b e r n we can write S = Ui~=lS{ where # (S i ) = # ( S ) / n , and the Si are pairwise disjoint.

P R O O F . The previous l emma shows tha t we can find S 1 in S with ~*($I) = I~(S) /n .

Apply this again to S \ $1 to construct $2, and so on. []

10.14 L E M M A . Let S be a finite, non-atomic measure space, and let 1 <_ p < co.

I f x , , . . . , xn E Lp( S) and e > O, then there exist s imple funct ions g l , . . . , g,~ o f the form

m

gi = E Aiy cj j : l

w h e r e {Ix, - ~,IIp < ~, ~ c ~ = 0 g ~ # / ~ , E j : ~ cj = 1, a ~ d f ~ j = ,(s)/,~.

122

P R O O F . Since the set of simple functions is dense in Lv(S), there exist simple functions

h i , . . . , h,~ such tha t Ilxi - hi]Iv < e/2. By intersecting the measurable sets from which the

hi are constructed, we can assume that these functions have the form

M

hi = E a l j u i y=I

where uy is the characteristic function of a set Ej . The sets E j can be assumed to form a

part i t ion of S. Choose N so that

Ne p >>_ 2Pmax laijt M#(S). i

j = l

By 10.12, each E j can be wri t ten as Ej = AjU Ba. , where A j A B i = [2, #(Bj) < p(S)/N,

and #(Aj) is a multiple of #(S)/N. By 10.12, each Ay can be wri t ten in the form

r(j)

As = U Djk k = l

where the sets Dyi, D y 2 , . . . , Djr(j) form a part i t ion of Aj and have measure #(S)/N.

Define B = UjM__iBy. Since M

B = S \ U A j j = l

we see tha t /~ (B) is a multiple of #(S)/N. By 10.11, B can be wri t ten as

~(o)

B = U Dok k = l

where the sets D0k form a part i t ion of B and have measure #(S)/N. Thus the sets

D:k (0 < 3" <- M, 1 < k < r(j)) part i t ion S into disjoint sets of measure #(S)/N.

Now define simple functions

M ~(J)

gi = Z Z fliik d, "k j = 0 k = 1

where djk is the characteristic function of Djk, and

3 i sk = o j = o.

123

Now

llh, - g, ll~ =

M M r(3.) p

- - 3 . = 0 k= l

M r(s) M M r(3.) p

s = O t = l v = l 3"=0 k = l

M r ( s ) M ~is t Pdst :zzflz°,.o. s : O t = l v----1

i15 .. s = l = = v=l

M p

B v - - 1

< f lai"lu~ B v = i

_<j @,°,.,)" = , ( B ) c~w

k v = l

= { M I ~ ( S ) / N } 1~,~1 <_ , , / z , .

It follows tha t

I I x , - g, llp ~ [ i x , - h, llp + [ [ h , - g, ll~ ~ ~, m

10 .15 L E M M A . Let z e Lp(S × T), where 1 < p < c~ and S and T are finite non-

atomic measure spaces. For each e > 0 there is a simple function f such that ]lz - - f l i p < e

and such that f b of the form

f = ~ i ajk cj dk 3.'~1 k = l

where c3. a n d dk are characterJstk functions of partitions S a nd T into sets of equal

measure.

P R O O F . By 1.52, there exist x i , . . . , x n 6 Lv(S ) and Y i , . . . , Y , E Lp(T) such tha t

Z n P -E~,®y, <,12. i = 1

124

By 10.14, there exist simple functions gl , . . . , g,~ in Lp(S) such that

lly, ll~ I I~, - g, ll~ < ~/4~

and each gi has a representation

m

giCs) = ~ a,, c,(s) 3 ' = 1

in which cj ¢k = 0 for j # k, Ecj = 1, and f cj(s)ds = I~(S)/m. Similarly, we find simple

functions hi E Lp (T) such that

and each hi has a representation

IIg~11~ I l y i - h, ll~ < e/4n

r

hi(t) = E aik dk k = l

where the characteristic functions dk have the same properties as the cj. We put

r~

f = E gi ® hi. i = l

Then

N z - ]Np <- II z - Exi @ yiHp + ]lExi ® yi - Eg, ® Y, llv + HEgl ® yl - Eg¢ ® hillp

< ~/2 + ~ l l ~ i - g, llp I ly,[lp + ~ l lg i l l~ ILy, - h i l l . < ~.

Now the form of f is

f = ) ' i ic j ® aikdk i=l \ y = l k=l

3"=1 k = l

125

C H A P T E R 11

A P P E N D I X O N M I S C E L L A N E O U S R E S U L T S I N B A N A C H S P A C E S

11.1 L E M M A . /.f P : X ~U and Q : X - - ~ V are Banach space projections such

that P Q P = QP, then the Boolean sum

P @ Q = P + Q - P Q

is a projection of X onto U -k V. The latter is therefore closed.

P R O O F . It is evident that P @ Q maps X into U ÷ V. In order to verify that it leaves

invariant each element of U, it suffices to write

(P @ Q ) P = p~ + Q P - P Q P = P + Q P - Q P = P.

Similarly, P $ Q leaves invariant each element of V, since

( p @ Q ) Q = p Q + Q 2 _ p Q 2 = p Q + Q _ p Q = Q . |

11.2 T H E O R E M . Let P be a projection of a Banach space X onto a subspace G,

and Q a projection of a Banach space Y onto a subspace H. Let ~ be a uniform norm on

X ® Y. Then

(P ®~ I) • ( I ®~ Q)

is a projection of X ®a Y onto

G ® Y + X ~ H .

The latter is therefore closed in X ®~ Y.

P R O O F . The operator P ® I is a projection of X ® Y onto G ® Y. By the uniformity

of c~, tiP ® Ill = [[PH- The continuous extension, P ®~ I , is a projection of X ®~ Y onto

G ~ Y . Similarly I ®~ Q is a projection of X ®a Y onto X ~ H . By using dyads, linearity,

and continuity, we can verify that

(P ®~ I ) ( I ® , Q) = (I ®~ Q ) ( P ® , I) .

The result then follows from 11.1. •

11.3 T H E O R E M . For a pair o f closed subspaces U and V in a Banach space, these

s ix proper t ies are equivalent:

(i) U + V is closed.

(ii) There is a cons tan t c such tha t each w E U + V can be represen ted as w = u + v,

wi th ,~ • u , ,, • v , and II~ll + t1~11 < cllwlI.

(ili) u I + v ± = (U n V ) ' .

(iv) U ± + V ± is weak*-closed.

(v) U ± + V ± is prox iminal .

(vi) U ± + V ± is dosed.

P R O O F . (i) implies (ii) [149, p. 1301. I n t r o d u c e a n o r m on U x g by writing IL(~,~)II =

I1~11 + IIvLI- Define a linear map L : U x V -4 U + V by writing L ( u , v ) = u + v . The map L

is continuous and surjective. Since U + V is complete, the open-mapping theorem applies

[149, p. 48]. Hence for some c and for each w • U + V there is an element (u, v) • U x V

such that L ( u , v ) -= w and tl(u,v)ll < cIIwil.

(ii) implies (iii) [108, p. 390]. The inclusion U ± + Y ± c (U N V) ± is elementary. For

the reverse inclusion, let ~ • ( U A V ) ±. Define ¢ on U + V as follows. If w • U + V , write

w = u + v w i t h u • U , v • V , andltul l+NvN <_cllwII. P u t ¢ ( w ) = ~ ( v ) . T h e d e f i n i t i o n i s

proper, for i f w = u I + v r then u ~ - u = v - v ~,whence v - v e • U N V a n d ~ ( v - v I) = 0 .

Also, ¢ is continuous since

I ¢ (~ ) I = i ~ ( , ) l ~ li~l111'4 < ~l[~il flail.

Let X be the Banach space in which U and V are situated. By the Hahn-Banach Theorem,

there exists 0 E X* such that 0 I (U + V) = ¢. Then 0 E U ± since

e(~) = ¢ (u ) = ¢ ( u + 0) = ¢(0).

If v E V then

( ~ - 0 ) ( v ) = ~ ( v ) - ¢ ( v ) = ~ ( v ) - ~ ( v ) = 0 .

Hence ~ o - 0 E V ±, and ~ = 0 + (~o-0) E U ± + V ±.

(iii) ==* (iv). The annihilator of any subspace of X is weak*-closed in X*.

(iv) :=~ (v). Any weak*-closed set in X* is proximinal [104, p. 123].

(v) ==* (vi). Any proximinM set is closed.

(vi) ===~ (i) [144, p. 555]. Define a linear map L : U --* X / V by writing L u = u + V.

Let

Q : X---, x / v a n d Q' : X* ---, X*/U"-

127

be the quotient maps. Let

A : (X/V)* -+ V ~ and B : X*/U-t -~ U*

be the canonical isometries given by A¢(x) = ¢ (x ÷ V) and B(~o + U-t) --- ~ [ U. One can

then verify that L* = BQeA and that the range of L* is B[U-t ÷ V-t]. Since U ± + V-t is

assumed closed, so is the range of L*. By [57, p. 488], the range of L is closed. This last

space is U+V, as a subspace of X/V. Finally, U + V is closed in X since it is Q - I [ U +V]. •

11.4 L E M M A . Let U and V be weak*-closed subspaces in a conjugate Banach

space X*. If U q- V is norm-dosed, then it is weak*-dosed and proximinal.

P R O O F . Since U and V are weak*-closed, they obey the equations U - (U-t)-t and

V -- (V-t)-t [108, p. 233]. Here we have put

for all ¢ E U } .

By 11.3, the desired conclusion follows. •

11.5 L E M M A . Let G be a tlnite-dimensionalsubspace with basis { g l , . . . , g,~} in a

Banach space X. Let Y be another Banach space, and a a reasonable crossnorm on X ® Y.

Then there exist bounded linear maps Li : X ®~ Y --* Y such that

z = ~ g i ® L i z for all z E G ® Y . i=1

P R O O I ~ . Select ~ol , . . . ,~,~ E X* so that ~oi(gy) = 6~y. Define Li on the (uncompleted)

tensor product X ® Y by

L i ( Z . x i ® y y ) = ~ p i ( x y ) y i ( l < i < n ) .

Notice that if the element z -- ~ y xy ® Yi is interpreted as a member of ~(X*, Y), the

defining equation for Li says, in effect, Liz = z(~i). Thus the definition of Liz is inde-

pendent of the representation of z. Since A(z) is the norm of z as an operator, we have,

by 1.6,

I]L.iz]l = I lz(~,)[I _< ACz)l[~,[[ ~ ~(z)[ [~,[ i .

This shows that

t lL, ii = sup{llL, li : _< 1} < tl ,Jl.

The operator Li is extended by continuity to X ®~ ]I, with the same bound [165, p. 39].

If z E G ® Y then

z = ~- 'gy®yy y = l

128

for appropr ia te Y5 c Y. Hence

L~z = z ( ~ ) = ~ ( g 5 ) Y 5 = Y," • y--1

11 .6 I, E M M A . Let G and H be ~nite-dimensional subspaces in Banach spaces X

and Y respectively. Let { g l , . . . , gu} and { h i , . . . , hm} be bases for G and g respectively.

Let a be a reasonable crossnorm on X ® Y. Then there exist bounded tinear operators

~ I : X ® ~ Y - - * Y and g y : X ® ~ Y - - * X

such that for each w E W = G ® Y + X ® H,

w = ~ g , ® ~ w + ~5 w @ h 5. i = l 5 = 1

P R O O F . By 11.2, there exist project ions

P : X ® , ~ Y - - ~ G ® Y and Q : X ® , Y - - - ~ X ® H

such tha t P + Q - Q P is a project ion of X ® , Y onto iV.

By 11.5 there exist bounde d linear maps

L ~ : X ® ~ Y - - - * Y and K 5 : X ® a Y ~ X

such tha t n

z = ~ gi ® Liz (z @ G ® Y ) i = l

m

z = g s z ® h5 e z ® H). 1--1

Define ~i = L i P and gy = K s Q ( I - P). Then for any w e W,

w = Pw + Q ( I - P )w

= ~ g , ® L , P w + ~ K s Q ( I - P ) w ® h 5 i----1 5=1

-'-= ~ gi ® ~iw + ~ ~jw ® h 5. • i = I 3'-----I

11.7 L E M M A . l I f E C(S x T) then the map s ~-~ fs from S to C(T) is continuous.

Hence the set {]s : s E S} is equicontinuous in C(T) .

P R O O F . Let so be any point of S. Given e > 0, we will find a ne ighborhood A/ of so

such tha t [[/s - fsol[ < e whenever s E A/. Thus we must determine 9V so tha t

[ f t (s) - f f (so) l < e (s e X, t e T) .

129

Since f is continuous at each point (So, t), there exist neighborhoods U(t) of so and

V ( t ) of t such that

I f ( o , ~ ) - f(so, t)l < ~/2 w henever (~ ,~) e U(t) x V(t).

By compactness of T, we find t l , . . . , t n such that V ( t l ) , . . . , V ( t , ) cover T. Define

= N u(t~). j : l

This neighborhood of so has the property desired, for if s E A/, if t E T, and if t E V(ti),

then (s,t) and (s0,t) belong to V( t , ) × Y ( t , ) , whence

I f t ( s ) - f t ( so ) l <_ I f ( s , t ) - f ( s o , t , ) l + I f ( s o , t , ) - f ( s o , t ) l < ~.

Since the map s H f s is continuous and S is compact, the set { f s : s E S ) is compact

in C ( T ) . Hence it is closed, bounded, and equicontinuous by the Ascoli Theorem. •

11.8 L E M M A . Let f be a cont inuous m a p o f a Banach space X into a Banach space

Y. I f T is a compac t Hausdor f f space then the m a p defined by ¢ ( z ) = f o z is cont inuous

f rom C ( T , X ) to C(T , Y ) .

P R O O F . It is obvious that ¢ maps C ( T , X ) into C(T , Y ) . For the continuity of this

map we proceed by contradiction. Suppose that there is a net za ~ zo such that

ll f o z ~ - f ozo ll ~ e .

For each a, select ta E T such that

I I f ( z ~ ( t ~ , ) ) - f (~o(t ,~) ) l l _> ~.

Take a convergent subnet, t~ ~ to. By the continuity of f at zo(to), there exists 5 > 0

such that for all x E X,

llx -- zo(to)ll < 5 ==~ l l f (x) -- f (zo( to)) l I < e/2.

By the continuity of zo at to, there is an index fl such that

l l z o ( t e ) - ~o(to)ll < 512 and I l z e - ~oI1 < 512.

Then 5 5

I I ~ ( t ~ ) - ~o(to)lt < l lz~(t~) - Zo(t~) l l + 1]zo(t~) - ~oCto)ll < 5- + ~"

130

Hence

[[f(za(t#)) -- f(zo(to))[[ < e /2 and Hf(zo(t#)) - f(zo(to))[I < e/2.

It follows f rom these two inequali t ies t ha t

I l f (z~( t , ) ) - f (zo(t~)) l l < ,,

which is a contradic t ion . •

11 .9 C O R O L L A R Y . If f is a continuous map of C(S) into itself, then the map f

de~ned by ( / z ) ( s , t) = ( f ( z t ) ) ( s ) is continuous from cCS x T) into itself.

P R O O F . Let J be the canonical i somet ry of C ( S x T) onto C(T, C(S) ) defined by

(Jz)( t ) = z t (z e C ( S x T)).

Let @ be the m a p p i n g of C(T, C(S)) to C(T, C(S)) defined by

4 . = / o , , ,~ e C(T , C ( S ) ) .

By 11.8, (I) is cont inuous. Since f = j - l ~ j , f is continuous. •

11 .10 L E M M A . Let S be a compact Hausdorff space. Let Iz be a reguIar Bore1

measu re such Shat I~(S) < oo. If w e n l (S) and f xw : 0 for all x • C(S) then w = O.

P R O O F . Since sgn w • Loo(S), we can find by Lus in ' s T h e o r e m [148, p. 56] a sequence

{x,~} in C(S) such t h a t [[x,,[[oo < 1 and ff

A . = {~ : x . ( . ) ¢ sgn ~(~)}

then #(A,~) --4 0. Now

Since ~ ( A . ) --~ 0, it follows from 10.10 that f a r [~J -~ 0 and so ~ = 0 •

11 .11 L E M M A . (Auerbach) I f X is an n-dimensional Banach space, then there

exist x l , . . . , x n in X and ¢ 1 , - - . , ¢ , ~ in X* such that

Nx~[[ = I[¢i]] = 1 and ¢ i (x i ) = ~ i j for l < i , j _ < n.

131

P R O O F . Fix any convenient basis {y i , . . . , y,~} for X. Select ¢ 1 , . . . , ¢,~ in the unit cell

of X* to maximize the determinant of the matrix A whose elements are Aij = ¢i(yj). Let

B denote the matrix inverse of A, and define xj = ~ = 1 B~.j Yr. Then

¢,(xi) = ¢, By1 yv := B , i ¢, (y . ) = B~j A, , = 5,3.

If )~ = det(A) then the cofactor of A~y is A Bj~. Let ¢ be a functional of norm I. If we

substitute ¢ for ¢i in the definition of A, the determinant is not increased. Computing

this determinant, using the elements in the ith row and their cofactors, gives us

~ ' ~ 1

It follows that ¢(xi) < 1. Since ¢ was arbitrary, IIx, II _ 1. Since ¢,(x,) = 1, Ilxill = 1. •

11.12 T H E O R E M . Let A be a compact linear operator from a Banach space X into

a Banach space Y. Let J r be the canor~ical embedding of Y into Y**. Then A**(X**) c

P R O O F . We assert that if X** and Y** are given their weak*-topologies then A** is a

continuous mapping from X** into Y**. Indeed, if {p~} is a net in X** with weak*-limit

0 then, for ¢ E Y*, we have

l im(A**p,)(¢) = l imp , (A*¢) = 0.

Now take p E X**. By the Goldstine Theorem [57, p. 424] we can find a net {Pa} in J x ( Z )

whose weak*-limit is p and [[p, ll < I]pII- Now for any ¢ E Y*, we have

(A**p)(¢) = lim(A**p~)(¢) = l i m ¢ ( A J x l p a ) .

By the compactness of A, { A J x l p a } contains a subnet {Axe} which converges in Y to

an element y. Hence

(A**p) (¢ ) = = ¢ ( y ) =

and so A** p = Jy(y) . •

11.13 D E F I N I T I O N . Let • be a set-valued mapping, taking points oYa topological

space S into the family of all subsets of a topological space T. The mapping • is said to

be lower semicontinuous if, for each open set 0 in T, the set

{s e s : ¢ (8) n 0 o }

132

is open in S.

11.1,1 T H E O R E M . (Michael Selection Theorem) Let • be a lower semicontinuous

map of a paracompact space S into the family of nonvoid closed convex subsets of a Banach

space X. Then ¢ has a continuous selection; i.e., there exists a continuous map ~ : S ~ X

such that ~(s) e ¢(s) for all s e S.

11.15 D E F I N I T I O N . Let (I) be a set-valued mapping, taking each point of a mea-

surable space S into a subset of a topological space T. We say that • is weakly measurable

if(I)- (0) is measurable in S whenever 0 is open in T. Here we have put, for any A c T,

¢ - ( A ) = {s e S : ¢(s) N A # El}.

11.16 T H E O R E M . (Kuratowski-Ryll-Nardzewski Measurable Selection Theorem)

Let t~ be a weakly measurable set-valued map which carries each point of a measurable

space S to a closed nonvoid subset of a complete separable metric space. Then • has a

measurable selection; i.e., there exists a function f : S ---* T such that f(s) C ¢(s) for

each s E S and f - l ( O ) is measurable for each open set 0 in T.

11.17 T H E O R E M . Let ¢ be a set-valued map, carrying each point of a measurable

space S to a closed nonvoid subset of a l~nite-dimensional Banach space X. If ~ - (K) is

measurable for each compact K in X then • has a measurable selection.

Let {xl, x2 , . . .} be a countable dense set in X. Consider the family of closed P R O O F .

cells

c . .~ = {~ : I lx- x.[I _< 1/m } (~,m e IN).

I t is easily seen that each open set 0 in X can be expressed as

0 = u { c , , , , : c , ~ c 0 } .

Since each Cnm is compact, (I)-(Cnm) is measurable. Hence (I)-(O) is measurable, since

¢ - ( 0 ) = u { m - ( c . m ) : c . m c o }

This proves that (I) is weakly measurable. An application of 11.16 completes the proof. ]]

133

N O T E S A N D R E M A R K S

C H A P T E R 1. The material of Chapter 1 is accessible in a number of sources. We

refer the reader to various monographs and textbooks rather than to original sources. An

early, but very readable, t reatment is Schatten's monograph [154]. Chapter 9 in Diestel

and Uhl [55] is also helpful. Among textbooks which treat the tensor product of linear

topological spaces, are those of Treves [165], Day [41], Khthe [113], and Schaefer [153].

The article by Gilbert and Leih [67] contains a succinct account of tensor-product theory.

The norms c~p discussed in 1.43 to 1.46 were introduced independently by Saphax

[152] and Chevet [36]. The paper of Saphar [151] contains a wealth of information about

them. The proof of the triangle inequality in 1.46 is from [151]. The isomorphism theorem

1.52 was apparently obtained independently by Chevet [35] and Persson [140].

C H A P T E R 2. The first half of Chapter 2 deals with the proximinality of G ® Y in

X ®a Y, where X and Y are Banach spaces, G is a subspace of X, and ~ is a crossnorm.

Results 2.1 to 2.3 and 2.5 to 2.7 are from Franchetti and Cheney [61]. The distance formula

in C(S, Y) (2.4) was given, along with many other important results, by Buck [25]. The

part of 2.4 which refers to ~oo (S, Y) is from von Golitschek and Cheney [76]. The results

2.10 to 2.13, concerning proximinality in spaces LI(S,Y), are contained implicitly or

explicitly in Khalil [112]. The result 2.13 generalizes work in [121], where techniques like

those in 2.8 gave a weaker result. An alternative approach to some of these proximinality

questions is via the theory of Chebyshev centers. These ideas are explored in Franchetti

and Cheney [62]. For recent results in the abstract theory of Chebyshev centers, the paper

of Amir, Mach, and Saatkamp [1] is recommended.

In the second half of Chapter 2, the theme is the proximinality of subspaces X ¢~ H +

G ~ Y in X ®a Y- Results 2.14 to 2.18 are from Respess and Cheney [146]. Corollary

2.19 was given first in [121]. Results 2.21 to 2.25 are also from [1461. Theorem 2.22 has

been called the "Sitting Duck Theorem" because it seems likely that its hypotheses can

be drastically weakened. The situation in L1 (S × T) is much more satisfactory, as can be

seen by comparing 2.22 with 2.26. The latter is from Holland, Light, and Sulley [102].

The problem of characterizing best approximations in spaces G ® Y q- X ® H has not

received much attention. The general theorems of Singer [160] axe of course applicable.

Havinson [99] proved a characterization theorem for best approximations in C(S × T) by

elements of the subspace C(S)÷ C(T). In [120], there is a characterization theorem similar

to Havinson's for subspaces C(S) ® H + G ® C(T), where G and H are finite dimensional.

Approximation in C(D), where D is a proper subset of S × T, differs in essential

aspects from appproximation in S × T. As 2.27 shows, the proximinality of C(S) + C(T) in C(D) depends on the geometry of D. The reader should consult the papers of Ofman

[137] and von Golitschek [69] for further information.

Theorem 2.22 requires a proximity map of C(S) satisfying a Lipschitz condition.

Unfortunately, this is a ra ther restrictive hypothesis, as the subspaces which are usually

encountered in practical problems do not have such proximity maps. For example, Haar

subspaces of dimension 2 or greater never have this property. Indeed, subspaces having

the Haar property at a single point do not have Lipschitz proximity maps. See Respess

and Cheney [145]. One can further prove that the proximity map onto a Haar subspace

of dimension 2 or greater is not uniformly continuous on bounded sets. By a theorem

of Freud, the proximity map of C(S) onto a Haar subspace is locally Lipschitz, but this

property seems to be inadequate for the proof of theorem 2.22. In general, C(S) will

contain subspaces of any dimension (finite or infinite) which possess Lipschitz proximity

maps [145].

The proximinality of spaces C(S) ® IIm(T) + I I , (S ) ® C(T), where S and T are

intervals and Hn is the space of polynomials of degree at most n, is an open question,

except when rain(n, rn) is 0. Von Golitschek and Cheney in [76] prove that each function

in C(S × T) possesses a best approximation in ~ (S) ® H1 (T) + H1 (S} ® ~oo (T) which is

continuous on the interior of the rectangle S × T.

C H A P T E R 3. According to Deutsch [51], the alternating algorithm appeared first

in mimeographed lecture notes of von Neumann in 1933. These notes were later published

in [135]. References to subsequent work on this Mgorithm for Hilbert space projections

will be found in Deutsch [49]. Applications of the algorithm to diverse problems, such as

the multigrid method for partial differential equations and to computerized tomography,

are mentioned in Deutsch [48]. Result 3.4 is from Franchetti and Light [63]. Item 3.5

is from [51]. The theorem in 3.6 derives from a paper by Baillon, Bruck and Reich [19];

i tem 3.7 is from [63]. The theorem in 3.8 is of course von Neumann's original result [135].

Results 3.9 and 3.10 are from [51]; the sequence 3.11 to 3.16, from [63].

Franchetti and Light in [63] have described a class of uniformly convex Banach spaces

in which the alternating algorithm is effective for any pair of closed subspaces. This class

properly includes the Hilbert spaces.

C H A P T E R 4. The concept of a central proximity map comes from Colomb's work

[78]. Items 4.1, 4.2, and 4.3 axe from [78], although the ideas are present in Diliberto and

Straus [56]. Item 4.7 is also from [78[.

C H A P T E R 5. Lemmas 5.1 and 5.2 are implicit in Diliberto and Straus [56]. Items

135

5.3 and 5.4 are from Aumann [6]. Item 5.6 comes from [56], but the proof is due to

Aumann [6]. The result 5.8 derives from [6], but the proof is from Light and Cheney

[122] It was proved by Dyn [58] that the Diliberto-Straus algorithm will fail if applied to

the subspaces C(S) ® III(T) and II0(S) ® C(T). Her method of proof was used by von

Golitschek and Cheney [75] to prove that the algorithm fails for C(S) ® H and G ® C(T)

whenever G and H are Haar subspaces such that dim G > 1 and dim H > 0. There exist

non-Haar subspaces G and H of arbi trary dimension, however, for which the algorithm

does work. See Respess and Cheney [145].

The convergence of the Diliberto-Straus algorithm can be arbitrarily slow, as is shown

by examples in yon Golitschek and Cheney [77]. An application of the algorithm to the

scaling (or "preconditioning") of matrices in the numerical solution of partial differential

equations can be found in several papers of Bank [20~21]. Often, one or two steps of the

algorithm suffice to produce a satisfactory preconditioning. The theoretical basis for this

is discussed in [77], where it is shown, for example, that the third and fourth iterates in

the algorithm satisfy the inequalities llz3 II <- 2 dist (z, W) and Hzhtl < 1.5 dist (z, W).

C H A P T E R 6. The results in this chapter are due to yon Golitschek in a series of

papers [69-74], especially [69]. For simplicity of exposition, we have limited the discussion

to domains S × T, although the original work includes certain types of closed subsets in

S × T .

C H A P T E R 7. The results 7.1 to 7.12 derive from Light and Holland [123]. Example

7.13 is from Light, McCabe, Phillips, and Cheney [124]. Lemma 7.15 comes from [124].

Items 7.16 to 7.19 are from {117] and [118].

C H A P T E R 8. Result 8.2 is due to Jameson and Pinkus [109]. Theorem 8.8 is from

Halton and Light [89]. Results 8.13 and 8.14 come from Franchett i and Cheney I60], and

are based upon the work in [109}. We thank Dr. Shekhtman for communicating Example

8.15 to us.

C H A P T E R 9. Theorem 9.1 is due to Rudin [150]. The techniques used in 9.2 were

kindly communicated to us by Y. Benyamini. Corollary 9.4 is from Halton and Light [90].

Theorem 9.5 comes from Halton and Light [88], while results 9.7 to 9.9 are from [90].

It is possible to obtain Theorem 9.9 directly from 9.1 without using the discretization

involved in 9.2 to 9.8. Details of this approach are contained in a preprint by W.A.Light

entitled "Minimal Projections in Tensor Product Spaces".

136

C H A P T E R 10. The material on the Bochner integral can be found in Diestel and

Vhl [55].

C H A P T E R 11. Lemma 11.1 on the Boolean sum of two projections is probably due

to Gordon [81]; see also Gordon and Cheney [84]. Theorem 11.2 is from Respess and Ch-

eney [146]. In Theorem 11.3, the equivalence of (i) and (ii) occurs in Rudin [149, p. 130].

The equivalence of (i) and (vi) is due to Reiter [144]. The proof of Auerbach's Lemma is

given by A.F. Ruston in ~Auerbach's lemma and tensor products of Banach spaces," Proc.

Cambridge Phil. Soc. 58 (1964), 476-480. The Michael Selection Theorem appeared first

in Michael [130]. See also his [131], Holmes' book [103], and Parthasarathy 's survey

[139]. The Kuratowski-Ryll-Nardzewski Measurable Selection Theorem first appeared in

their paper [114]. See also [139], Wagner's survey update [166], and Diestel's book [54,

p. 268].

137

B I O G R A P H I C A L S K E T C H O F R O B E R T S C H A T T E N

Robert Schatten was born in LwSw, Poland, on January 28, 1911. He received the

Magister degree from John Casimir University (LwSw) in 1933. After emigrat ing to the

United States, he enrolled in the graduate school of Columbia University, receiving an

M.A. degree in 1939. He continued his research under the direction of Francis J. Murray,

and was awarded the Ph.D. degree in 1942. He held a brief appoin tment as a lecturer in

the College of Pha rmacy at Columbia University in 1942 before joining the U.S. Army,

in which he served from 1942 to 1943. He suffered a broken back during training at Fort

Benning, Georgia, and this injury gave him much pain for the rest of his llfe. In the

academic year 1943-1944, he had an appointment as assistant professor at the University

of Vermont. He then won a two-year appointment as a Fellow of the Nat ional Research

Council, and divided his t ime during this period between the Inst i tute for Advanced Study

and Yale University. He collaborated during these years with John yon Neamann and with

Nelson Dunford. In 1946, Schatten began a long association with the University of Kansas,

first as associate professor (1946-1952) and then as professor (1952-1961). This tenure was

interrupted by leaves in 1950 and in 1952-1953, bo th of which he spent at the Inst i tute for

Advanced Study. The year 1960-1961 was spent as a visiting professor at the University

of Southern California, and in 1961-1962 he served as professor at the State University

of New York at Stony Brook. In 1962 he became professor at Hunter College, where he

remained until his death on August 26, 1977. During the years 1964-1972 he was also

a member of the doctoral faculty of the Graduate School of the City University of New

York. At the t ime of his death there were no immediate survivors, all his known relatives

in Poland having been killed during the war.

To his former students, Schatten will be remembered as a dedicated teacher who was

genuinely concerned with the intellectual development of his students. They will certainly

not forget his unique style of lecturing. He always spoke without a book or notes, and

rarely used the blackboard. His lectures were extremely clear and well-organized; he

never lost his way in complicated arguments. The pace was such that the students could

(and were expected to) take notes verbat im; if they did so, their notes would read like a

polished book, except for some linguistic idiosyncracies such as, ~Given is a set...". He

left nothing to chance in his dictation; for example, he invariably ended an argument with

~This concludes the proof."

Schatten had his own way of making abstract concepts memorable to his elementary

classes. Who could forget what a s e q u e n c e was after hearing Schatten describe a long

corridor, stretching as far as the eye could see, with hooks regularly spaced on the wall

and numbered 1,2, 3 , . . . ? "Then," Schatten would say, "I come along with a big bag of

numbers over my shoulder, and hang one number on each hook." This of course was

accompanied by suitable gestures for emphasis.

Schatten had some eccentricities which endeared him to his friends. He hated noise,

especially when it interrupted his sleep. In Lawrence, Kansas, he was seen early one

morning in his garden, clad in pyjamas, trying to shoo away the grackles from a tree near

his bedroom. Cars were also his b~tes noires: al though he owned a car at one time, he

never fully mastered the art of driving. He once got a nasty bruise from a t tempt ing to

put his head out of the car window before lowering the glass. Bachelor life also presented

various pitfalls such as having to contend with laundries that insisted on ironing his socks.

He kept his unpublished mathemat ica l researches in a bank 's safe-deposit box.

Schat ten 's principal mathemat ica l achievement was tha t of initiating the s tudy of

tensor products of Banach spaces. The concepts of crossnorm, associate norm, greatest

crossnorm, least crossnorm, and uniform crossnorm, all either originated with him or at

least first received careful s tudy in his papers. He was mainly interested in the applications

of this subject to linear t ransformations on Hilbert space. In this subject, the "Schatten

Classes" perpe tua te his name.

P u b l i c a t i o n s o f R o b e r t S c h a t t e n

1. On the direct product of Banaeh spaces, Trans. Amer. Math. Soc. 53 (1943), 195-217.

MR 4-161.

2. On reflexive norms for the direct product of Banach spaces, Trans. Amer. Math. Soc.

54 (1943}, 498-506. MR 5-99.

3. (with N. Dunford), On the associate and conjugate space for the direct product of

Banach spaces, Trans. Amer. Math. Soc. 59 (1946), 430-436. MR 7-455.

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152

I N D E X O F N O T A T I O N

7

ker(A)

[ ' ( X , Y }

X *

U ±

®

®~

®~

®

® v~ v

J_

( , )

A

0

A

V

J~CA)

IR

iv,

XE

generic crossnorm 4

Hilbert space crossnorm 22

least of the reasonable crossnorms 3

greatest crossnorm 6

p-nuclear cross norm 27

kernel of the operator A 41

space of bounded linear maps from X into Y 8

conjugate of the Banach space X 1

annihilator of the subspace U 41

Boolean sum 25

tensor product (uncompleted} 1

completion of tensor product with norm ~ 9

completion of tensor product with norm "~ 12

completion of tensor product with norm c~ 9

closure of tensor product subspace 12

completion of tensor product with norm c~p 29

completion of tensor product with norm fl 22

orthogonal symbol 25

inner product 20

point functional: ~(x) = x(~) 100, 72

composition of two functions 10

minimum in lattice sense 68

maximum in lattice sense 68

range of the operator A 26

real line 1

set of positive integers 1

characteristic function of the set E 108, 113

[]

Lp(S)

c(s)

Lp(S, X)

L o(S)

dist

X8

X t

a*

c(s,Y)

IIAII. ~ ( X , Y * )

A ® B

A ® ~ B

Dq

6

incl(U, v) /]A

Z)

Jx

H

A,

empty set

surjective map

end of proof

Lebesgue space on S

space of continuous functions on S

space of Lp-maps from S to X

essentially bounded functions on S

distance functional

bounded functions on S

s-section of x : x~(t) = x ( s , t )

t-section of x : x~(s) = x(s, t)

tensor

dual of the norm a

continuous functions from S to Y

a-norm of an operator

operators of bounded a-norm

space of pth power summable sequences

tensor product of operators

continuous extension of A ® B

special functional

modulus of convexity

inclination of two subspaces

restriction of function f to the set A

projection constant of W in Z

canonical imbedding of X in X**

mapping indicator

adjoint of an operator

36

35

2

11, 26, 24

9

11, 29

34

36

36

11

11

1

4

9

14

14

19

19

2O

26

49

54

12

91

17

10, 30, 25

17

154

/ r

IR ~

A \ B

K

S × T

sgn

z)

identity operator on space Y

n-dimensional real space

l~ n normed with the p-norm

set difference

bounded maps from S into X

closure of the set K

Cartesian product of two sets

averaging functional

signum function

relative projection constant

25

27

27

10

37

48

10

60

80

91

155

I N D E X

Algebraic tensor product, 2

Algorithm

alternating, 48

Diliberto-Straus, 60, 62

von Golitschek, 67

Approximation property, 17

Associate norm, 4

Atom, 121

Auerbach lemma, 131

Averaging functional, 60

E-norm, 22

Best approximation, 35

Biorthonormal, 46

Bochner

integral, 113

integrable function, 117, 11

Boolean sum, 25,126

Central proximity map, 56

Chebyshev subspace, 35, 36, 49

Commuting with G, 103

Completed tensor product, 9

Convexity, modulus of, 49

Crossnorm, 3, 4

reasonable, 4

uniform, 8

associate, 4

greatest, 6, 7

Dyad, 3

Diliberto-Straus algorithm, 60, 62

Equivalent expressions, 1

Expressions, 1

Four-point rule, 104

"~-norm, 6

yon Colitschek algorithm, 67

Colomb's theory, 56

Greatest crossnorm, 6

Haar subspace, 36

Inclination, 54

Injective norm, 35

Invariant subspace under a group, 103

James' theorem, 80

Kuratowski-Ryll-Nardzewski theorem, 133

h-norm, 3

Least of reasonable norms, 3

Measurable selection theorem, 133

Measurable set, 113

Michael's theorem, 132

Minimal projection, 91

Modulus of convexity, 49

Nonatomic, 121

Nonexpansive, 51, 61

Norm, 3

p-nuclear, 27

a, 4

ap, 27

/~, 22

7, 6

A, 3

Null set, 113

Orthogonal projection, 59

P-nuclear norm, 27

Path, 69

Projections

minimal, 91,103

orthogonal, 25

Projection constant, 91

Property B, 97

Proximinal, 35

Proximity map, 35, 48

Reasonable norm, 4

Schatten, R., 138

Section of a function, 11, 60

Selection theorem

Michael's 133, 37

Kuratowski-Ryll- Nardzewski, 133

Simple function, 11,113

Sitting-Duck theorem, 44, 134

Smooth

point, 48

space, 48

subspace, 54

Strongly measurable, 113

Tensor product, 1

algebraic, 2

completed, 9

of Hilbert spaces, 20

of operators, 19

Uniform crossnorm, 8

Uniformly convex, 50

Von Neuman algorithm, 48,60

Weakly measurable, 38

157