TijaniPakhrou - downloads.hindawi.comdownloads.hindawi.com/journals/jfs/2017/7347130.pdf ·...
5
Research Article Best -Simultaneous Approximation in (, ) Tijani Pakhrou Department of Mathematics, Faculty of Sciences, University of Alicante, 03080 Alicante, Spain Correspondence should be addressed to Tijani Pakhrou; [email protected] Received 6 April 2017; Accepted 21 May 2017; Published 12 July 2017 Academic Editor: Yoshihiro Sawano Copyright © 2017 Tijani Pakhrou. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let be a Banach space. Let 1≤<∞ and denote by (, ) the Banach space of all -valued Bochner -integrable functions on a certain positive complete -finite measure space (Ω, Σ, ), endowed with the usual -norm. In this paper, the theory of liſting is used to prove that, for any weakly compact subset of , the set (, ) is -simultaneously proximinal in (, ) for any arbitrary monotonous norm in R . Dedicated to Hirak Chaabi Rife˜ no 1. Introduction roughout this paper, (, ‖ ⋅ ‖) is a real Banach space and ∗ is the closed unit ball of ∗ , the dual of , with ( ∗ , )- topology. Let (Ω, Σ, ) be a positive complete -finite measure space; is a liſting and is the associated liſting topology (a base for this topology is {() \ : ∈ Σ, is a null set}; see [1, p. 59]). For an ∈ and ∗ ∈ ∗ , ∗ () will also be denoted by ⟨ ∗ , ⟩. Let 1≤<∞. We denote by (, ) the Banach space of all Bochner -integrable functions on Ω with values in endowed with the usual norm = (∫ Ω () ) 1/ (1) for every ∈ (, ) (see [2]). Let be a positive integer. We say that a norm in R is monotonous if, for every = ( ) 1≤≤ , = ( ) 1≤≤ ∈ R such that | | ≤ | | for =1,...,; we have () ≤ () . (2) Let be a subset of ; we say that 0 ∈ is a best –simultaneous approximation from of the vectors 1 ,..., ∈ if ( 1 − 0 ,..., − 0 ) ≤( 1 − ,..., − ), (3) for every ∈. If every -tuple of vectors 1 ,..., ∈ admits a best –simultaneous approximation from , then is said to be –simultaneously proximinal in . Of course, for =1, the preceding concepts are just best approximation and proximinality. e problem of the best -simultaneous approximation in (, ) has been deeply and extensively studied; see, for example, [3–11]. When is a reflexive subspace of , it was proved in [3] that (, ) is –simultaneously proximinal in (, ), where is the Lebesgue measure on = [0, 1]. We have also obtained similar results, in the Banach space 1 ( 0 , ) of -valued Bochner 0 -integrable functions defined on Ω, where 0 is the restriction of to a certain sub- -algebra Σ 0 of Σ and is a reflexive space (see [8]). Our purpose is to study the -simultaneous proximinal- ity of the set (, ) defined by (, ) fl { ∈ (, ) : () ∈ for a.e. ∈ Ω} , (4) for certain subsets of . 2. Preliminaries In this section we wish to include some facts which are fundamental in our study. Hindawi Journal of Function Spaces Volume 2017, Article ID 7347130, 4 pages https://doi.org/10.1155/2017/7347130
Transcript of TijaniPakhrou - downloads.hindawi.comdownloads.hindawi.com/journals/jfs/2017/7347130.pdf ·...
Research ArticleBest119873-Simultaneous Approximation in 119871119901(120583119883)
Tijani Pakhrou
Department of Mathematics Faculty of Sciences University of Alicante 03080 Alicante Spain
Correspondence should be addressed to Tijani Pakhrou tijanipakhrouuaes
Received 6 April 2017 Accepted 21 May 2017 Published 12 July 2017
Academic Editor Yoshihiro Sawano
Copyright copy 2017 Tijani Pakhrou This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Let 119883 be a Banach space Let 1 le 119901 lt infin and denote by 119871119901(120583 119883) the Banach space of all 119883-valued Bochner 119901-integrable functionson a certain positive complete 120590-finite measure space (Ω Σ 120583) endowed with the usual 119901-norm In this paper the theory of liftingis used to prove that for any weakly compact subset 119882 of 119883 the set 119871119901(120583 119882) is 119873-simultaneously proximinal in 119871119901(120583 119883) for anyarbitrary monotonous norm 119873 in R119899
Dedicated to Hirak Chaabi Rifeno
1 Introduction
Throughout this paper (119883 sdot ) is a real Banach space and119861119883lowast is the closed unit ball of119883lowast the dual of119883 with120590(119883lowast 119883)-topology Let (Ω Σ 120583) be a positive complete120590-finitemeasurespace 120588 is a lifting and 120591120588 is the associated lifting topology (abase for this topology is 120588(119860) 119861 119860 isin Σ 119861 is a null setsee [1 p 59]) For an 119909 isin 119883 and 119909lowast isin 119883lowast 119909lowast(119909) will also bedenoted by ⟨119909lowast 119909⟩
Let 1 le 119901 lt infin We denote by 119871119901(120583 119883) the Banach spaceof all Bochner 119901-integrable functions on Ω with values in 119883endowed with the usual norm
10038171003817100381710038171198911003817100381710038171003817119901 = (intΩ
1003817100381710038171003817119891 (119904)1003817100381710038171003817119901 119889120583)1119901 (1)
for every 119891 isin 119871119901(120583 119883) (see [2])Let 119899 be a positive integer We say that a norm 119873 inR119899 is
monotonous if for every 119905 = (119905119894)1le119894le119899 119904 = (119904119894)1le119894le119899 isin R119899 suchthat |119905119894| le |119904119894| for 119894 = 1 119899 we have
119873 (119905) le 119873 (119904) (2)Let 119884 be a subset of 119883 we say that 1199100 isin 119884 is a
best 119873ndashsimultaneous approximation from 119884 of the vectors1199091 119909119899 isin 119883 if119873 (10038171003817100381710038171199091 minus 11991001003817100381710038171003817 1003817100381710038171003817119909119899 minus 11991001003817100381710038171003817)
le 119873 (10038171003817100381710038171199091 minus 1199101003817100381710038171003817 1003817100381710038171003817119909119899 minus 1199101003817100381710038171003817) (3)
for every 119910 isin 119884 If every 119899-tuple of vectors 1199091 119909119899 isin 119883admits a best 119873ndashsimultaneous approximation from 119884 then119884 is said to be 119873ndashsimultaneously proximinal in 119883 Of coursefor 119899 = 1 the preceding concepts are just best approximationand proximinality
The problem of the best 119873-simultaneous approximationin 119871119901(120583 119883) has been deeply and extensively studied see forexample [3ndash11] When 119884 is a reflexive subspace of 119883 it wasproved in [3] that 119871119901(120583 119884) is 119873ndashsimultaneously proximinalin 119871119901(120583 119883) where 120583 is the Lebesgue measure on 119868 =[0 1] We have also obtained similar results in the Banachspace 1198711(1205830 119883) of119883-valued Bochner 1205830-integrable functionsdefined on Ω where 1205830 is the restriction of 120583 to a certain sub-120590-algebra Σ0 of Σ and 119883 is a reflexive space (see [8])
Our purpose is to study the 119873-simultaneous proximinal-ity of the set 119871119901(120583 119883) defined by
119871119901 (120583 119882)fl 119892 isin 119871119901 (120583 119883) 119892 (119904) isin 119882 for ae 119904 isin Ω (4)
for certain subsets 119882 of 119883
2 Preliminaries
In this section we wish to include some facts which arefundamental in our study
HindawiJournal of Function SpacesVolume 2017 Article ID 7347130 4 pageshttpsdoiorg10115520177347130
2 Journal of Function Spaces
Definition 1 Let 1199091 119909119899 be vectors in 119883 one says that asequence (119910119896)119896ge1 in119884 sub 119883 is119873-simultaneously approximatingto 1199091 119909119899 in 119884 if
lim119896rarrinfin
119873 (10038171003817100381710038171199091 minus 1199101198961003817100381710038171003817 1003817100381710038171003817119909119899 minus 1199101198961003817100381710038171003817)= inf119911isin119884
119873 (10038171003817100381710038171199091 minus 1199111003817100381710038171003817 1003817100381710038171003817119909119899 minus 1199111003817100381710038171003817) (5)
Lemma 2 ([8 Lemma 1]) Let 1199091 119909119899 be vectors in 119883 andlet (119910119896)119896ge1 be a 119873-simultaneously approximating sequence to1199091 119909119899 in 119884 sub 119883 Assume that (119910119896)119896ge1 is weakly convergentto 1199100 isin 119884 Then 1199100 is a best 119873-simultaneous approximationfrom 119884 of 1199091 119909119899
The next result gives a property of 119873-simultaneouslyapproximating sequences in the space 1198711(120583 119883)The followinglemma is taken from [8 Lemma 2] which was proved for thecase when 119884 = 119883 and Σ0 = ΣLemma 3 ([8 Lemma 2]) Let 119884 be a subset of 119883 Let1198911 119891119899 be functions in 1198711(120583 119883) and let (119892119896)119896ge1 sub 1198711(120583 119884)be an 119873-simultaneously approximating sequence to 1198911 119891119899in 1198711(120583 119884) If (119860119896)119896ge1 is a sequence of Σ-measurable setssuch that lim119896rarrinfin120583(119860119896) = 0 then (119892119896120594119860119888
119896)119896ge1 is also an119873-simultaneously approximating sequence to 1198911 119891119899 in1198711(120583 119884)
We also need a result which is probably the best knownldquosubsequence splitting lemmardquo in integrable function spaces
Lemma 4 (Kadec-Pełczynski-Rosenthal Lemma 213 in[12]) If (119891119898)119898ge1 is a bounded sequence in 1198711(120583) then thereexists a subsequence (119891119898119896)119896ge1 of (119891119898)119898ge1 and a sequence(119860119896)119896ge1 of pairwise disjoint measurable sets such that(119891119898119896120594119860119888119896)119896ge1 is uniformly integrable
3 Main Results
In this section we give the simple and direct proofs of themain results These results are similar to [6] However in[6] more hypotheses are necessary and the proofs are totallydifferent
Theorem 5 Let 119882 be a weakly compact subset of 119883 Then1198711(120583 119882) is 119873ndashsimultaneously proximinal in 1198711(120583 119883)Proof Let 1198911 119891119899 be functions in 1198711(120583 119883) and let(ℎ119898)119898ge1 sub 1198711(120583 119882) be a 119873-simultaneously approximatingsequence to 1198911 119891119899 in 1198711(120583 119882) We have
lim119898rarrinfin
119873 (10038171003817100381710038171198911 minus ℎ11989810038171003817100381710038171 1003817100381710038171003817119891119899 minus ℎ11989810038171003817100381710038171)= infℎisin1198711(120583119882)
119873 (10038171003817100381710038171198911 minus ℎ10038171003817100381710038171 1003817100381710038171003817119891119899 minus ℎ10038171003817100381710038171) (6)
Since119873(1198911minusℎ1198981 119891119899minusℎ1198981) is a convergent sequenceby taking into account the fact that for each 119898 ge 1 we have
119873 (1 1) 1003817100381710038171003817ℎ11989810038171003817100381710038171 le 119873 (10038171003817100381710038171198911 minus ℎ11989810038171003817100381710038171 1003817100381710038171003817119891119899 minus ℎ11989810038171003817100381710038171)+ 119873 (1003817100381710038171003817119891110038171003817100381710038171 100381710038171003817100381711989111989910038171003817100381710038171) (7)
one deduces that (ℎ119898)119898ge1 is bounded in 1198711(120583 119882)
By Lemma 4 there exists a subsequence (ℎ119898119896)119896ge1 of(ℎ119898)119898ge1 and a sequence (119860119896)119896ge1 of pairwise disjoint mea-surable sets such that (ℎ119898119896120594119860119888119896)119896ge1 is uniformly integrable in1198711(120583 119882)
On the one handinfinsum119896=1
120583 (119860119896) = 120583 (infin⋃119896=1
119860119896) le 120583 (Ω) lt infin (8)
so we have lim119896rarrinfin120583(119860119896) = 0 Therefore by Lemma 3(ℎ119898119896120594119860119888119896)119897ge1 is a119873-simultaneously approximating sequence to1198911 119891119899 in 1198711(120583 119882)Let us denote 119892119896 = ℎ119898119896120594119860119888119896 for each 119896 ge 1Since 119882 sub 119883 is weakly compact then the sequence(119892119896(119904))119896ge1 sub 119882 for ae 119904 isin Ω and has a weakly convergent
subsequence which again is denoted by (119892119896(119904))119896ge1 Let usdenote by 119892(119904) its weak limit for ae 119904 isin Ω Therefore for each119909lowast isin 119883lowast the numerical function 119909lowast(119892) is 120583-measurable So 119892is weakly120583-measurable On the other hand for each 119896 isin N119892119896is 120583-essentially separably valued that is there exists 119860119896 isin Σwith 120583(119860119896) = 0 and such that 119892119896(Ω 119860119896) is a norm separablesubset of 119883 For each 119896 let us pick a dense and countablesubset 119863119896 of 119892119896(Ω 119860119896) Then the set
119884 = span(infin⋃119896=1
119863119896) (9)
is norm closed and separable For every 119896 isin N and 119904 isin Ω 119860119896we have 119892119896(119904) isin 119884 Since 119892(119904) is weak limit of (119892119896(119904))119896ge1 forae 119904 isin Ω we obtain that 119892(119904) isin 119884 for ae 119904 isin Ω Thus 119892 is 120583-essentially separably valued Therefore Pettis MeasurabilityTheorem [2 p 42] guarantees that the function 119892 Ω rarr 119883 is120583-measurable
Since (119892119896(119904))119896ge1 is weakly convergent to 119892(119904) isin 119882 for ae119904 isin Ω then (119892119896(119904))119896ge1 is bounded and1003817100381710038171003817119892 (119904)1003817100381710038171003817 le lim inf
119896rarrinfin
1003817100381710038171003817119892119896 (119904)1003817100381710038171003817 (10)
for ae 119904 isin Ω Using this result the boundedness of (119892119896)119896ge1and Fatoursquos lemma [13 Theorem 283 p 131] we get 119892 isin1198711(120583 119882)
We claim that 119892 is in the closed convex hull of (119892119896)119896ge1in (1198711(120583 119883) sdot 1) Suppose this is not true Then by [14Theorem 92 p 65] there is 119906 isin (1198711(120583 119883))lowast such that
⟨119906 119892⟩ lt inf119896ge1
⟨119906 119892119896⟩ (11)
Now we will use the result that (1198711(120583 119883))lowast is thespace of all bounded continuous mapping 119906 (Ω 120591120588) rarr(119883lowast 120590(119883lowast 119883)) (see [1 Theorem 7 p 94]) we get
lim119896rarrinfin
119906 (119904) (119892119896 (119904)) = 119906 (119904) (119892 (119904)) (12)
ae on Ω Using Egoroff rsquos theorem [13 Theorem 221 p 110]and uniform integrability of (119892119896)119896ge1 we get ⟨119906 119892119896⟩ 997888997888997888997888rarr
119896rarrinfin⟨119906 119892⟩ a contradiction By taking convex combinations of
elements of (119892119896)119896ge1 if necessary we can assume that 119892119896 sdot1997888997888997888997888rarr119896rarrinfin
Journal of Function Spaces 3
119892 in 1198711(120583 119882)Therefore by Lemma 2 we have the fact that 119892is the best 119873-simultaneous approximation from 1198711(120583 119882) of1198911 119891119899
Although the proof of the following theorem is similar tothat of Theorem 5 we provide the proof here as a means forthe reader to readily justify these assertions
Theorem 6 Let 119882 be a weakly compact subset of 119883 Then119871119901(120583 119882) is119873ndashsimultaneously proximinal in 119871119901(120583 119883) for each1 lt 119901 lt infin
Proof Let 1198911 119891119899 be functions in 119871119901(120583 119883) and let(119892119896)119896ge1 sub 119871119901(120583 119882) be a 119873-simultaneously approximatingsequence to 1198911 119891119899 in 119871119901(120583 119882) We have
lim119896rarrinfin
119873 (10038171003817100381710038171198911 minus 1198921198961003817100381710038171003817119901 1003817100381710038171003817119891119899 minus 1198921198961003817100381710038171003817119901)= infℎisin119871119901(120583119882)
119873 (10038171003817100381710038171198911 minus ℎ1003817100381710038171003817119901 1003817100381710038171003817119891119899 minus ℎ1003817100381710038171003817119901) (13)
Notice that for each 119896 ge 1 we have
119873 (1 1) 10038171003817100381710038171198921198961003817100381710038171003817119901 le 119873 (10038171003817100381710038171198911 minus 1198921198961003817100381710038171003817119901 1003817100381710038171003817119891119899 minus 1198921198961003817100381710038171003817119901)+ 119873 (100381710038171003817100381711989111003817100381710038171003817119901 10038171003817100381710038171198911198991003817100381710038171003817119901) (14)
Since 119873(1198911 minus 119892119896119901 119891119899 minus 119892119896119901) is a convergent sequencewe deduce that (119892119896)119896ge1 is bounded in 119871119901(120583 119882) Therefore(119892119896)119896ge1 is uniformly integrable
Since 119882 sub 119883 is weakly compact then the sequence(119892119896(119904))119896ge1 sub 119882 for ae 119904 isin Ω has a weakly convergentsubsequence which again is denoted by (119892119896(119904))119896ge1 Let usdenote by 119892(119904) its weak limit for ae 119904 isin Ω Therefore for each119909lowast isin 119883lowast the numerical function 119909lowast(119892) is 120583-measurable So 119892is weakly120583-measurable On the other hand for each 119896 isin N119892119896is 120583-essentially separably valued that is there exists 119860119896 isin Σwith 120583(119860119896) = 0 and such that 119892119896(Ω 119860119896) is a norm separablesubset of 119883 For each 119896 let us pick a dense and countablesubset 119863119896 of 119892119896(Ω 119860119896) Then the set
119884 = span(infin⋃119896=1
119863119896) (15)
is norm closed and separable For every 119896 isin N and 119904 isin Ω 119860119896we have 119892119896(119904) isin 119884 Since 119892(119904) is weak limit of (119892119896(119904))119896ge1for ae 119904 isin Ω we obtain that 119892(119904) isin 119884 for ae 119904 isin ΩThus 119892 is 120583-essentially separably valued Therefore the PettisMeasurabilityTheorem [2 p 42] guarantees that the function119892 Ω rarr 119883 is 120583-measurable
Since (119892119896(119904))119896ge1 is weakly convergent to 119892(119904) isin 119882 for ae119904 isin Ω then (119892119896(119904))119896ge1 is bounded and
1003817100381710038171003817119892 (119904)1003817100381710038171003817 le lim inf119896rarrinfin
1003817100381710038171003817119892119896 (119904)1003817100381710038171003817 (16)
for ae 119904 isin Ω Using this result the boundedness of (119892119896)119896ge1and Fatoursquos lemma [13 Theorem 283 p 131] we get 119892 isin119871119901(120583 119882)
We claim that 119892 is in the closed convex hull of (119892119896)119896ge1in (119871119901(120583 119883) sdot 119901) Suppose this is not true Then by [14Theorem 92 p 65] there is a 119906 isin (119871119901(120583 119883))lowast such that
⟨119906 119892⟩ lt inf119896ge1
⟨119906 119892119896⟩ (17)
Now we will be using the result that (119871119901(120583 119883))lowastis the space of all bounded continuous mapping 119906 (Ω 120591120588) rarr (119883lowast 120590(119883lowast 119883)) (see [1 Theorem 9 p 97])we get lim119896rarrinfin119906(119904)(119892119896(119904)) = 119906(119904)(119892(119904)) ae on Ω UsingEgoroff rsquos theorem [13 Theorem 221 p 110] and uniformintegrability of (119892119896)119896ge1 we get ⟨119906 119892119896⟩ 997888997888997888997888rarr
119896rarrinfin⟨119906 119892⟩ a
contradiction By taking convex combinations of elements
of (119892119896)119896ge1 if necessary we can assume that 119892119896 sdot119901997888997888997888997888rarr119896rarrinfin
119892 in119871119901(120583 119882) Therefore by Lemma 2 we have the fact that 119892is the best 119873-simultaneous approximation from 119871119901(120583 119882) of119891119901 119891119899
Let Σ0 be a subndash120590ndashalgebra of Σ and 1205830 the restriction of 120583to Σ0 Let 1 le 119901 lt infin We defined the set
119871119901 (1205830 119882)fl 119892 isin 119871119901 (1205830 119883) 119892 (119904) isin 119882 for ae 119904 isin Ω (18)
where119882 is a subset of119883We obtain analogously the followingtheorem
Theorem 7 Let 119882 be a weakly compact subset of 119883 and 1 le119901 lt infin Then 119871119901(1205830 119882) is 119873ndashsimultaneously proximinal in119871119901(1205830 119883)Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Research is partially supported byMTM 2012-31286 (SpanishMinistry of Economy and Competitiveness)
References
[1] A Ionescu Tulcea and C Ionescu Tulcea Topics in theTheory ofLiftings Spring Berlin Germany 1969
[2] J Diestel and J J Uhl ldquoVectorMeasuresrdquoMathematical SurveysandMonographs AmericanMathematical Society vol 15 articleRI 1977
[3] F B Saidi D Hussein and R Khalil ldquoBest simultaneousapproximation in 119871119901(119868 119864)rdquo Journal of Approximation Theoryvol 116 no 2 pp 369ndash379 2002
[4] A L Garkavi ldquoOn the Chebyshev center and the convex hullof a setrdquo Uspekhi Matematicheskikh Nauk vol 19 pp 139ndash1451964
[5] S V Konyagin ldquoA remark on renormings of nonreflexive spacesand the existence of a Chebyshev centerrdquo Moscow UniversityMathematics Bulletin vol 43 no 2 pp 55-56 1988
4 Journal of Function Spaces
[6] X-F Luo C Li H-K Xu and J-C Yao ldquoExistence ofbest simultaneous approximations in 119871119901(119878 Σ 119883)rdquo Journal ofApproximation Theory vol 163 no 9 pp 1300ndash1316 2011
[7] J Mach ldquoBest simultaneous approximation of bounded func-tions with values in certain Banach spacesrdquo MathematischeAnnalen vol 240 no 2 pp 157ndash164 1979
[8] J Mendoza and T Pakhrou ldquoBest simultaneous approximationin 1198711(120583 119883)rdquo Journal of ApproximationTheory vol 145 no 2 pp212ndash220 2007
[9] T S Rao ldquoApproximation properties for spaces of Bochnerintegrable functionsrdquo Journal of Mathematical Analysis andApplications vol 423 no 2 pp 1540ndash1545 2015
[10] J Shi and R Huotari ldquoSimultaneous approximation fromconvex setsrdquo Computers amp Mathematics with Applications AnInternational Journal vol 30 no 3-6 pp 197ndash206 1995
[11] L Vesely ldquoGeneralized centers of finite sets in Banach spacesrdquoActa Mathematica Universitatis Comenianae (NS) vol 66 pp83ndash115 1997
[12] P Cembranos and J Mendoza ldquoBanach spaces of vector-valuedfunctionsrdquo in Lecture Notes in Mathematics vol 1676 SpringerBerlin Germany 1997
[13] V I Bogachev Measure Theory vol I Springer Berlin Ger-many 2007
[14] H H Schaeffer Topological Vector Spaces Springer BerlinGermany 1986
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces
Definition 1 Let 1199091 119909119899 be vectors in 119883 one says that asequence (119910119896)119896ge1 in119884 sub 119883 is119873-simultaneously approximatingto 1199091 119909119899 in 119884 if
lim119896rarrinfin
119873 (10038171003817100381710038171199091 minus 1199101198961003817100381710038171003817 1003817100381710038171003817119909119899 minus 1199101198961003817100381710038171003817)= inf119911isin119884
119873 (10038171003817100381710038171199091 minus 1199111003817100381710038171003817 1003817100381710038171003817119909119899 minus 1199111003817100381710038171003817) (5)
Lemma 2 ([8 Lemma 1]) Let 1199091 119909119899 be vectors in 119883 andlet (119910119896)119896ge1 be a 119873-simultaneously approximating sequence to1199091 119909119899 in 119884 sub 119883 Assume that (119910119896)119896ge1 is weakly convergentto 1199100 isin 119884 Then 1199100 is a best 119873-simultaneous approximationfrom 119884 of 1199091 119909119899
The next result gives a property of 119873-simultaneouslyapproximating sequences in the space 1198711(120583 119883)The followinglemma is taken from [8 Lemma 2] which was proved for thecase when 119884 = 119883 and Σ0 = ΣLemma 3 ([8 Lemma 2]) Let 119884 be a subset of 119883 Let1198911 119891119899 be functions in 1198711(120583 119883) and let (119892119896)119896ge1 sub 1198711(120583 119884)be an 119873-simultaneously approximating sequence to 1198911 119891119899in 1198711(120583 119884) If (119860119896)119896ge1 is a sequence of Σ-measurable setssuch that lim119896rarrinfin120583(119860119896) = 0 then (119892119896120594119860119888
119896)119896ge1 is also an119873-simultaneously approximating sequence to 1198911 119891119899 in1198711(120583 119884)
We also need a result which is probably the best knownldquosubsequence splitting lemmardquo in integrable function spaces
Lemma 4 (Kadec-Pełczynski-Rosenthal Lemma 213 in[12]) If (119891119898)119898ge1 is a bounded sequence in 1198711(120583) then thereexists a subsequence (119891119898119896)119896ge1 of (119891119898)119898ge1 and a sequence(119860119896)119896ge1 of pairwise disjoint measurable sets such that(119891119898119896120594119860119888119896)119896ge1 is uniformly integrable
3 Main Results
In this section we give the simple and direct proofs of themain results These results are similar to [6] However in[6] more hypotheses are necessary and the proofs are totallydifferent
Theorem 5 Let 119882 be a weakly compact subset of 119883 Then1198711(120583 119882) is 119873ndashsimultaneously proximinal in 1198711(120583 119883)Proof Let 1198911 119891119899 be functions in 1198711(120583 119883) and let(ℎ119898)119898ge1 sub 1198711(120583 119882) be a 119873-simultaneously approximatingsequence to 1198911 119891119899 in 1198711(120583 119882) We have
lim119898rarrinfin
119873 (10038171003817100381710038171198911 minus ℎ11989810038171003817100381710038171 1003817100381710038171003817119891119899 minus ℎ11989810038171003817100381710038171)= infℎisin1198711(120583119882)
119873 (10038171003817100381710038171198911 minus ℎ10038171003817100381710038171 1003817100381710038171003817119891119899 minus ℎ10038171003817100381710038171) (6)
Since119873(1198911minusℎ1198981 119891119899minusℎ1198981) is a convergent sequenceby taking into account the fact that for each 119898 ge 1 we have
119873 (1 1) 1003817100381710038171003817ℎ11989810038171003817100381710038171 le 119873 (10038171003817100381710038171198911 minus ℎ11989810038171003817100381710038171 1003817100381710038171003817119891119899 minus ℎ11989810038171003817100381710038171)+ 119873 (1003817100381710038171003817119891110038171003817100381710038171 100381710038171003817100381711989111989910038171003817100381710038171) (7)
one deduces that (ℎ119898)119898ge1 is bounded in 1198711(120583 119882)
By Lemma 4 there exists a subsequence (ℎ119898119896)119896ge1 of(ℎ119898)119898ge1 and a sequence (119860119896)119896ge1 of pairwise disjoint mea-surable sets such that (ℎ119898119896120594119860119888119896)119896ge1 is uniformly integrable in1198711(120583 119882)
On the one handinfinsum119896=1
120583 (119860119896) = 120583 (infin⋃119896=1
119860119896) le 120583 (Ω) lt infin (8)
so we have lim119896rarrinfin120583(119860119896) = 0 Therefore by Lemma 3(ℎ119898119896120594119860119888119896)119897ge1 is a119873-simultaneously approximating sequence to1198911 119891119899 in 1198711(120583 119882)Let us denote 119892119896 = ℎ119898119896120594119860119888119896 for each 119896 ge 1Since 119882 sub 119883 is weakly compact then the sequence(119892119896(119904))119896ge1 sub 119882 for ae 119904 isin Ω and has a weakly convergent
subsequence which again is denoted by (119892119896(119904))119896ge1 Let usdenote by 119892(119904) its weak limit for ae 119904 isin Ω Therefore for each119909lowast isin 119883lowast the numerical function 119909lowast(119892) is 120583-measurable So 119892is weakly120583-measurable On the other hand for each 119896 isin N119892119896is 120583-essentially separably valued that is there exists 119860119896 isin Σwith 120583(119860119896) = 0 and such that 119892119896(Ω 119860119896) is a norm separablesubset of 119883 For each 119896 let us pick a dense and countablesubset 119863119896 of 119892119896(Ω 119860119896) Then the set
119884 = span(infin⋃119896=1
119863119896) (9)
is norm closed and separable For every 119896 isin N and 119904 isin Ω 119860119896we have 119892119896(119904) isin 119884 Since 119892(119904) is weak limit of (119892119896(119904))119896ge1 forae 119904 isin Ω we obtain that 119892(119904) isin 119884 for ae 119904 isin Ω Thus 119892 is 120583-essentially separably valued Therefore Pettis MeasurabilityTheorem [2 p 42] guarantees that the function 119892 Ω rarr 119883 is120583-measurable
Since (119892119896(119904))119896ge1 is weakly convergent to 119892(119904) isin 119882 for ae119904 isin Ω then (119892119896(119904))119896ge1 is bounded and1003817100381710038171003817119892 (119904)1003817100381710038171003817 le lim inf
119896rarrinfin
1003817100381710038171003817119892119896 (119904)1003817100381710038171003817 (10)
for ae 119904 isin Ω Using this result the boundedness of (119892119896)119896ge1and Fatoursquos lemma [13 Theorem 283 p 131] we get 119892 isin1198711(120583 119882)
We claim that 119892 is in the closed convex hull of (119892119896)119896ge1in (1198711(120583 119883) sdot 1) Suppose this is not true Then by [14Theorem 92 p 65] there is 119906 isin (1198711(120583 119883))lowast such that
⟨119906 119892⟩ lt inf119896ge1
⟨119906 119892119896⟩ (11)
Now we will use the result that (1198711(120583 119883))lowast is thespace of all bounded continuous mapping 119906 (Ω 120591120588) rarr(119883lowast 120590(119883lowast 119883)) (see [1 Theorem 7 p 94]) we get
lim119896rarrinfin
119906 (119904) (119892119896 (119904)) = 119906 (119904) (119892 (119904)) (12)
ae on Ω Using Egoroff rsquos theorem [13 Theorem 221 p 110]and uniform integrability of (119892119896)119896ge1 we get ⟨119906 119892119896⟩ 997888997888997888997888rarr
119896rarrinfin⟨119906 119892⟩ a contradiction By taking convex combinations of
elements of (119892119896)119896ge1 if necessary we can assume that 119892119896 sdot1997888997888997888997888rarr119896rarrinfin
Journal of Function Spaces 3
119892 in 1198711(120583 119882)Therefore by Lemma 2 we have the fact that 119892is the best 119873-simultaneous approximation from 1198711(120583 119882) of1198911 119891119899
Although the proof of the following theorem is similar tothat of Theorem 5 we provide the proof here as a means forthe reader to readily justify these assertions
Theorem 6 Let 119882 be a weakly compact subset of 119883 Then119871119901(120583 119882) is119873ndashsimultaneously proximinal in 119871119901(120583 119883) for each1 lt 119901 lt infin
Proof Let 1198911 119891119899 be functions in 119871119901(120583 119883) and let(119892119896)119896ge1 sub 119871119901(120583 119882) be a 119873-simultaneously approximatingsequence to 1198911 119891119899 in 119871119901(120583 119882) We have
lim119896rarrinfin
119873 (10038171003817100381710038171198911 minus 1198921198961003817100381710038171003817119901 1003817100381710038171003817119891119899 minus 1198921198961003817100381710038171003817119901)= infℎisin119871119901(120583119882)
119873 (10038171003817100381710038171198911 minus ℎ1003817100381710038171003817119901 1003817100381710038171003817119891119899 minus ℎ1003817100381710038171003817119901) (13)
Notice that for each 119896 ge 1 we have
119873 (1 1) 10038171003817100381710038171198921198961003817100381710038171003817119901 le 119873 (10038171003817100381710038171198911 minus 1198921198961003817100381710038171003817119901 1003817100381710038171003817119891119899 minus 1198921198961003817100381710038171003817119901)+ 119873 (100381710038171003817100381711989111003817100381710038171003817119901 10038171003817100381710038171198911198991003817100381710038171003817119901) (14)
Since 119873(1198911 minus 119892119896119901 119891119899 minus 119892119896119901) is a convergent sequencewe deduce that (119892119896)119896ge1 is bounded in 119871119901(120583 119882) Therefore(119892119896)119896ge1 is uniformly integrable
Since 119882 sub 119883 is weakly compact then the sequence(119892119896(119904))119896ge1 sub 119882 for ae 119904 isin Ω has a weakly convergentsubsequence which again is denoted by (119892119896(119904))119896ge1 Let usdenote by 119892(119904) its weak limit for ae 119904 isin Ω Therefore for each119909lowast isin 119883lowast the numerical function 119909lowast(119892) is 120583-measurable So 119892is weakly120583-measurable On the other hand for each 119896 isin N119892119896is 120583-essentially separably valued that is there exists 119860119896 isin Σwith 120583(119860119896) = 0 and such that 119892119896(Ω 119860119896) is a norm separablesubset of 119883 For each 119896 let us pick a dense and countablesubset 119863119896 of 119892119896(Ω 119860119896) Then the set
119884 = span(infin⋃119896=1
119863119896) (15)
is norm closed and separable For every 119896 isin N and 119904 isin Ω 119860119896we have 119892119896(119904) isin 119884 Since 119892(119904) is weak limit of (119892119896(119904))119896ge1for ae 119904 isin Ω we obtain that 119892(119904) isin 119884 for ae 119904 isin ΩThus 119892 is 120583-essentially separably valued Therefore the PettisMeasurabilityTheorem [2 p 42] guarantees that the function119892 Ω rarr 119883 is 120583-measurable
Since (119892119896(119904))119896ge1 is weakly convergent to 119892(119904) isin 119882 for ae119904 isin Ω then (119892119896(119904))119896ge1 is bounded and
1003817100381710038171003817119892 (119904)1003817100381710038171003817 le lim inf119896rarrinfin
1003817100381710038171003817119892119896 (119904)1003817100381710038171003817 (16)
for ae 119904 isin Ω Using this result the boundedness of (119892119896)119896ge1and Fatoursquos lemma [13 Theorem 283 p 131] we get 119892 isin119871119901(120583 119882)
We claim that 119892 is in the closed convex hull of (119892119896)119896ge1in (119871119901(120583 119883) sdot 119901) Suppose this is not true Then by [14Theorem 92 p 65] there is a 119906 isin (119871119901(120583 119883))lowast such that
⟨119906 119892⟩ lt inf119896ge1
⟨119906 119892119896⟩ (17)
Now we will be using the result that (119871119901(120583 119883))lowastis the space of all bounded continuous mapping 119906 (Ω 120591120588) rarr (119883lowast 120590(119883lowast 119883)) (see [1 Theorem 9 p 97])we get lim119896rarrinfin119906(119904)(119892119896(119904)) = 119906(119904)(119892(119904)) ae on Ω UsingEgoroff rsquos theorem [13 Theorem 221 p 110] and uniformintegrability of (119892119896)119896ge1 we get ⟨119906 119892119896⟩ 997888997888997888997888rarr
119896rarrinfin⟨119906 119892⟩ a
contradiction By taking convex combinations of elements
of (119892119896)119896ge1 if necessary we can assume that 119892119896 sdot119901997888997888997888997888rarr119896rarrinfin
119892 in119871119901(120583 119882) Therefore by Lemma 2 we have the fact that 119892is the best 119873-simultaneous approximation from 119871119901(120583 119882) of119891119901 119891119899
Let Σ0 be a subndash120590ndashalgebra of Σ and 1205830 the restriction of 120583to Σ0 Let 1 le 119901 lt infin We defined the set
119871119901 (1205830 119882)fl 119892 isin 119871119901 (1205830 119883) 119892 (119904) isin 119882 for ae 119904 isin Ω (18)
where119882 is a subset of119883We obtain analogously the followingtheorem
Theorem 7 Let 119882 be a weakly compact subset of 119883 and 1 le119901 lt infin Then 119871119901(1205830 119882) is 119873ndashsimultaneously proximinal in119871119901(1205830 119883)Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Research is partially supported byMTM 2012-31286 (SpanishMinistry of Economy and Competitiveness)
References
[1] A Ionescu Tulcea and C Ionescu Tulcea Topics in theTheory ofLiftings Spring Berlin Germany 1969
[2] J Diestel and J J Uhl ldquoVectorMeasuresrdquoMathematical SurveysandMonographs AmericanMathematical Society vol 15 articleRI 1977
[3] F B Saidi D Hussein and R Khalil ldquoBest simultaneousapproximation in 119871119901(119868 119864)rdquo Journal of Approximation Theoryvol 116 no 2 pp 369ndash379 2002
[4] A L Garkavi ldquoOn the Chebyshev center and the convex hullof a setrdquo Uspekhi Matematicheskikh Nauk vol 19 pp 139ndash1451964
[5] S V Konyagin ldquoA remark on renormings of nonreflexive spacesand the existence of a Chebyshev centerrdquo Moscow UniversityMathematics Bulletin vol 43 no 2 pp 55-56 1988
4 Journal of Function Spaces
[6] X-F Luo C Li H-K Xu and J-C Yao ldquoExistence ofbest simultaneous approximations in 119871119901(119878 Σ 119883)rdquo Journal ofApproximation Theory vol 163 no 9 pp 1300ndash1316 2011
[7] J Mach ldquoBest simultaneous approximation of bounded func-tions with values in certain Banach spacesrdquo MathematischeAnnalen vol 240 no 2 pp 157ndash164 1979
[8] J Mendoza and T Pakhrou ldquoBest simultaneous approximationin 1198711(120583 119883)rdquo Journal of ApproximationTheory vol 145 no 2 pp212ndash220 2007
[9] T S Rao ldquoApproximation properties for spaces of Bochnerintegrable functionsrdquo Journal of Mathematical Analysis andApplications vol 423 no 2 pp 1540ndash1545 2015
[10] J Shi and R Huotari ldquoSimultaneous approximation fromconvex setsrdquo Computers amp Mathematics with Applications AnInternational Journal vol 30 no 3-6 pp 197ndash206 1995
[11] L Vesely ldquoGeneralized centers of finite sets in Banach spacesrdquoActa Mathematica Universitatis Comenianae (NS) vol 66 pp83ndash115 1997
[12] P Cembranos and J Mendoza ldquoBanach spaces of vector-valuedfunctionsrdquo in Lecture Notes in Mathematics vol 1676 SpringerBerlin Germany 1997
[13] V I Bogachev Measure Theory vol I Springer Berlin Ger-many 2007
[14] H H Schaeffer Topological Vector Spaces Springer BerlinGermany 1986
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 3
119892 in 1198711(120583 119882)Therefore by Lemma 2 we have the fact that 119892is the best 119873-simultaneous approximation from 1198711(120583 119882) of1198911 119891119899
Although the proof of the following theorem is similar tothat of Theorem 5 we provide the proof here as a means forthe reader to readily justify these assertions
Theorem 6 Let 119882 be a weakly compact subset of 119883 Then119871119901(120583 119882) is119873ndashsimultaneously proximinal in 119871119901(120583 119883) for each1 lt 119901 lt infin
Proof Let 1198911 119891119899 be functions in 119871119901(120583 119883) and let(119892119896)119896ge1 sub 119871119901(120583 119882) be a 119873-simultaneously approximatingsequence to 1198911 119891119899 in 119871119901(120583 119882) We have
lim119896rarrinfin
119873 (10038171003817100381710038171198911 minus 1198921198961003817100381710038171003817119901 1003817100381710038171003817119891119899 minus 1198921198961003817100381710038171003817119901)= infℎisin119871119901(120583119882)
119873 (10038171003817100381710038171198911 minus ℎ1003817100381710038171003817119901 1003817100381710038171003817119891119899 minus ℎ1003817100381710038171003817119901) (13)
Notice that for each 119896 ge 1 we have
119873 (1 1) 10038171003817100381710038171198921198961003817100381710038171003817119901 le 119873 (10038171003817100381710038171198911 minus 1198921198961003817100381710038171003817119901 1003817100381710038171003817119891119899 minus 1198921198961003817100381710038171003817119901)+ 119873 (100381710038171003817100381711989111003817100381710038171003817119901 10038171003817100381710038171198911198991003817100381710038171003817119901) (14)
Since 119873(1198911 minus 119892119896119901 119891119899 minus 119892119896119901) is a convergent sequencewe deduce that (119892119896)119896ge1 is bounded in 119871119901(120583 119882) Therefore(119892119896)119896ge1 is uniformly integrable
Since 119882 sub 119883 is weakly compact then the sequence(119892119896(119904))119896ge1 sub 119882 for ae 119904 isin Ω has a weakly convergentsubsequence which again is denoted by (119892119896(119904))119896ge1 Let usdenote by 119892(119904) its weak limit for ae 119904 isin Ω Therefore for each119909lowast isin 119883lowast the numerical function 119909lowast(119892) is 120583-measurable So 119892is weakly120583-measurable On the other hand for each 119896 isin N119892119896is 120583-essentially separably valued that is there exists 119860119896 isin Σwith 120583(119860119896) = 0 and such that 119892119896(Ω 119860119896) is a norm separablesubset of 119883 For each 119896 let us pick a dense and countablesubset 119863119896 of 119892119896(Ω 119860119896) Then the set
119884 = span(infin⋃119896=1
119863119896) (15)
is norm closed and separable For every 119896 isin N and 119904 isin Ω 119860119896we have 119892119896(119904) isin 119884 Since 119892(119904) is weak limit of (119892119896(119904))119896ge1for ae 119904 isin Ω we obtain that 119892(119904) isin 119884 for ae 119904 isin ΩThus 119892 is 120583-essentially separably valued Therefore the PettisMeasurabilityTheorem [2 p 42] guarantees that the function119892 Ω rarr 119883 is 120583-measurable
Since (119892119896(119904))119896ge1 is weakly convergent to 119892(119904) isin 119882 for ae119904 isin Ω then (119892119896(119904))119896ge1 is bounded and
1003817100381710038171003817119892 (119904)1003817100381710038171003817 le lim inf119896rarrinfin
1003817100381710038171003817119892119896 (119904)1003817100381710038171003817 (16)
for ae 119904 isin Ω Using this result the boundedness of (119892119896)119896ge1and Fatoursquos lemma [13 Theorem 283 p 131] we get 119892 isin119871119901(120583 119882)
We claim that 119892 is in the closed convex hull of (119892119896)119896ge1in (119871119901(120583 119883) sdot 119901) Suppose this is not true Then by [14Theorem 92 p 65] there is a 119906 isin (119871119901(120583 119883))lowast such that
⟨119906 119892⟩ lt inf119896ge1
⟨119906 119892119896⟩ (17)
Now we will be using the result that (119871119901(120583 119883))lowastis the space of all bounded continuous mapping 119906 (Ω 120591120588) rarr (119883lowast 120590(119883lowast 119883)) (see [1 Theorem 9 p 97])we get lim119896rarrinfin119906(119904)(119892119896(119904)) = 119906(119904)(119892(119904)) ae on Ω UsingEgoroff rsquos theorem [13 Theorem 221 p 110] and uniformintegrability of (119892119896)119896ge1 we get ⟨119906 119892119896⟩ 997888997888997888997888rarr
119896rarrinfin⟨119906 119892⟩ a
contradiction By taking convex combinations of elements
of (119892119896)119896ge1 if necessary we can assume that 119892119896 sdot119901997888997888997888997888rarr119896rarrinfin
119892 in119871119901(120583 119882) Therefore by Lemma 2 we have the fact that 119892is the best 119873-simultaneous approximation from 119871119901(120583 119882) of119891119901 119891119899
Let Σ0 be a subndash120590ndashalgebra of Σ and 1205830 the restriction of 120583to Σ0 Let 1 le 119901 lt infin We defined the set
119871119901 (1205830 119882)fl 119892 isin 119871119901 (1205830 119883) 119892 (119904) isin 119882 for ae 119904 isin Ω (18)
where119882 is a subset of119883We obtain analogously the followingtheorem
Theorem 7 Let 119882 be a weakly compact subset of 119883 and 1 le119901 lt infin Then 119871119901(1205830 119882) is 119873ndashsimultaneously proximinal in119871119901(1205830 119883)Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Research is partially supported byMTM 2012-31286 (SpanishMinistry of Economy and Competitiveness)
References
[1] A Ionescu Tulcea and C Ionescu Tulcea Topics in theTheory ofLiftings Spring Berlin Germany 1969
[2] J Diestel and J J Uhl ldquoVectorMeasuresrdquoMathematical SurveysandMonographs AmericanMathematical Society vol 15 articleRI 1977
[3] F B Saidi D Hussein and R Khalil ldquoBest simultaneousapproximation in 119871119901(119868 119864)rdquo Journal of Approximation Theoryvol 116 no 2 pp 369ndash379 2002
[4] A L Garkavi ldquoOn the Chebyshev center and the convex hullof a setrdquo Uspekhi Matematicheskikh Nauk vol 19 pp 139ndash1451964
[5] S V Konyagin ldquoA remark on renormings of nonreflexive spacesand the existence of a Chebyshev centerrdquo Moscow UniversityMathematics Bulletin vol 43 no 2 pp 55-56 1988
4 Journal of Function Spaces
[6] X-F Luo C Li H-K Xu and J-C Yao ldquoExistence ofbest simultaneous approximations in 119871119901(119878 Σ 119883)rdquo Journal ofApproximation Theory vol 163 no 9 pp 1300ndash1316 2011
[7] J Mach ldquoBest simultaneous approximation of bounded func-tions with values in certain Banach spacesrdquo MathematischeAnnalen vol 240 no 2 pp 157ndash164 1979
[8] J Mendoza and T Pakhrou ldquoBest simultaneous approximationin 1198711(120583 119883)rdquo Journal of ApproximationTheory vol 145 no 2 pp212ndash220 2007
[9] T S Rao ldquoApproximation properties for spaces of Bochnerintegrable functionsrdquo Journal of Mathematical Analysis andApplications vol 423 no 2 pp 1540ndash1545 2015
[10] J Shi and R Huotari ldquoSimultaneous approximation fromconvex setsrdquo Computers amp Mathematics with Applications AnInternational Journal vol 30 no 3-6 pp 197ndash206 1995
[11] L Vesely ldquoGeneralized centers of finite sets in Banach spacesrdquoActa Mathematica Universitatis Comenianae (NS) vol 66 pp83ndash115 1997
[12] P Cembranos and J Mendoza ldquoBanach spaces of vector-valuedfunctionsrdquo in Lecture Notes in Mathematics vol 1676 SpringerBerlin Germany 1997
[13] V I Bogachev Measure Theory vol I Springer Berlin Ger-many 2007
[14] H H Schaeffer Topological Vector Spaces Springer BerlinGermany 1986
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Function Spaces
[6] X-F Luo C Li H-K Xu and J-C Yao ldquoExistence ofbest simultaneous approximations in 119871119901(119878 Σ 119883)rdquo Journal ofApproximation Theory vol 163 no 9 pp 1300ndash1316 2011
[7] J Mach ldquoBest simultaneous approximation of bounded func-tions with values in certain Banach spacesrdquo MathematischeAnnalen vol 240 no 2 pp 157ndash164 1979
[8] J Mendoza and T Pakhrou ldquoBest simultaneous approximationin 1198711(120583 119883)rdquo Journal of ApproximationTheory vol 145 no 2 pp212ndash220 2007
[9] T S Rao ldquoApproximation properties for spaces of Bochnerintegrable functionsrdquo Journal of Mathematical Analysis andApplications vol 423 no 2 pp 1540ndash1545 2015
[10] J Shi and R Huotari ldquoSimultaneous approximation fromconvex setsrdquo Computers amp Mathematics with Applications AnInternational Journal vol 30 no 3-6 pp 197ndash206 1995
[11] L Vesely ldquoGeneralized centers of finite sets in Banach spacesrdquoActa Mathematica Universitatis Comenianae (NS) vol 66 pp83ndash115 1997
[12] P Cembranos and J Mendoza ldquoBanach spaces of vector-valuedfunctionsrdquo in Lecture Notes in Mathematics vol 1676 SpringerBerlin Germany 1997
[13] V I Bogachev Measure Theory vol I Springer Berlin Ger-many 2007
[14] H H Schaeffer Topological Vector Spaces Springer BerlinGermany 1986
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
