1: X too:= X Yn} X, - University of...

Problem 1: a) Check if the following spaces are metric spaces:i) X = too:= {(Xn)nEN: Xn E IR for each nand suplxnl < oo}. d(x,y) = sup{lxn-Ynl: n EN}.ii) X = foo, d(x,y) = #{n EN: xn #- Yn} (Hamming distance).iii) Take X to be London. For every pair of points x, y E X, let d(x, y) be the distance that acar needs to drive from x to y. (Taxicab metric, this is not the distance "as the crow flies").iv) Let X be a chess board, and for any pair of squares x, yE X, d(x, y) is the number of stepsthat a knight needs to go from x to y.

v) As iv, but now with a bishop.

b) The diameter of metric space (X, d) is sup{d(x, y) : x, y E X}. Find the diameters of theexamples given in Notes 1.c) Find the diameters of the examples in part a).

Problem 2: Show that for all x, y, z in a metric space (X, d): d(x, y) 2 Id(x, z) - d(z, y)l·

Problem 3: An (open) ball of radius r is B(x;r) = {y EX: d(x,y) < r}. A closed ballor radius r is B(x; r) = {y EX: d(x, y) ::; r}. Take X = IR2. Draw the open ball B(O; 1) (Le.,the unit ball) ifa) d is Euclidean metric.b) d is sum metric.c) d is sup metric.d) d is discrete metric.

Problem 4: a) If d1 and d2 a metrics, check if the following functions are also metrics:i) d1 + d2;

ii) max{d1, d2};

iii) min{d1, d2l;

iv) ~d1 + ~d2'v) d1 . d2.

b) For each of the four axioms in the definition of metric, find an example d that fails it butsatisfies the other three axioms.

Problem 5: Let X = 0([0,1]) and fn(t) = tn be a sequence in X.a) What is the limit in the sup metric?b) What is the limit in the L1 metric?c) Show that in any metric, a sequence can have at most one limit.

Problem 6: a) Let x = (2-n)nEN, y = «( -1)n2-n)nEN and z = (1,0,0,0, ... ).a) Compute IlxllI, Ily111,and IlzliI- Also compute d1(x, y), d1(x, z) and d1(y, z).

b) Compute Ilxllooand Ilzlloo'c) Compute IIxllp and Ilzllp verify what happens as p -+ 00.

Problem 7: a) For which rE IRis En ;r convergent?

b) For which P does (~)nEN belong to RP. And (jn)nEN?c) Let 1 :::;p < q ::; 00. Find a sequence x such that x E £q but x 1- £p.

d) Show that £ife £f1if ::; p < q ::; 00.

Problem 8: a) Check the triangle inequality for (£00,111100)and for (£1, 11lid·b) The triangle inequality also holds for (£P, 11lip) and pE [1,(0), but for pE (1,00) it is much

1harder to check. Can you find a counter-example to show that (£2,11111) is not a normed space?

2

Problem 9: a) Show that any Cauchy sequence can have at most one limit.b) A subsequence of a Cauchy sequence is Cauchy.c) Give an example of a non-Cauchy sequence with a Cauchy subsequence.

Problem 10: Let X = C([-I, 1]) with norm 11lit- Let (fn)nEN be defined by

fn(x) = { ~X-1if x E [~, I),ifxE(_ll) n' n '

if x E [-1, -~].

a) Show that (fn)nEN is Cauchy.b) Is (X, 11lid a Banach space?c) Is (fn)nEN convergent in (L1([-I, 1]),11 111)?

QI

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Problem 12: Let f be a contraction of the metric space (X, d).

a) Show that f can have at most one fixed point.b) Show that f is continuous.

Problem 11: Let lP'n([a, 1]) be the space of all polynomials p : [a, 1] ~ IR of degree::; nand

lP'([a,l]) be the space of alll1lynomials p : [a, 1] ~ IR of any degree.a) Is lP'n([a, 1]) a linear space" What is its dimension?b) Question a) but now for lP'([a, 1]).

c) Use the norm 111100,i.e. uniform convergence. Is lP'([a, 1]) a proper normed space with thisnorm?

d) Find a sequence (Pn)nEN that converges to f(x) = eX.

e) Is (lP'([a, 1]),11 11(0)a Banach space?f) Show that if Pn ~ 9 in 111100,then f is a continuous function.

"

I~Problem 13: Let X be the space of all Lipschitz functions f : [a, 1] ~ R / \:

a) Is X a linear space? / ~~

b) Is X a subspace of C([a, I])? Give an example of f E C([a, 1]) \ X. / ~ r

c) Is (X, 111100)a Banach space?

(') r,

(~ / (")''" \.\,/ '\'

~~"'- )~~~ ~ ~~

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Problem 14: Let (X, d) be a metric space.a) Show that a Lipschitz function is continuous.b) Fix P E X and define f : X --+ ~ by f (x) = d(p, x). Show that f a Lipschitz function. Whatis a smallest Lipschitz constant.c) Let d' on X X X defined by <tt(Xl, Yl), (X2, Y2)) = d(Xl, X2) + d(Yl, Y2). Show that d' is ametric.

d) Let F: X X X --+ ~ be given by F(x,y) ~d(x,y). Conclude that F is continuous.-

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