# BestN-Simultaneous Approximation in Lp(μ,X)

Authors: Tijani Pakhrou
Journal: Journal of Function Spaces
 (2017)

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## Abstract

LetX be a Banach space. Let 1≤p<∞ and denote by Lp(μ,X) the Banach space of all X-valued Bochner p-integrable functions on a certain positive complete σ-finite measure space (Ω,Σ,μ), endowed with the usual p-norm. In this paper, the theory of lifting is used to prove that, for any weakly compact subset W of X, the set Lp(μ,W) is N-simultaneously proximinal in Lp(μ,X) for any arbitrary monotonous norm N in Rn.


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## Body

## 1. Introduction

Throughout this paper,(X,·) is a real Banach space and BX∗ is the closed unit ball of X∗, the dual of X, with σ(X∗,X)-topology. Let (Ω,Σ,μ) be a positive complete σ-finite measure space; ρ is a lifting and τρ is the associated lifting topology (a base for this topology is ρ(A)∖B:A∈Σ,B is a null set; see [1, p. 59]). For an x∈X and x∗∈X∗, x∗(x) will also be denoted by 〈x∗,x〉.Let1≤p<∞. We denote by Lp(μ,X) the Banach space of all Bochner p-integrable functions on Ω with values in X endowed with the usual norm (1)fp=∫Ωfspdμ1/pfor every f∈Lp(μ,X) (see [2]).Letn be a positive integer. We say that a norm N in Rn is monotonous if, for every t=(ti)1≤i≤n, s=(si)1≤i≤n∈Rn such that |ti|≤|si| for i=1,…,n; we have (2)Nt≤Ns.LetY be a subset of X; we say that y0∈Y is a best N–simultaneous approximation fromYof the vectorsx1,…,xn∈X if (3)Nx1-y0,…,xn-y0≤Nx1-y,…,xn-y,for every y∈Y. If every n-tuple of vectors x1,…,xn∈X admits a best N–simultaneous approximation from Y, then Y is said to be N–simultaneously proximinal inX. Of course, for n=1, the preceding concepts are just best approximation and proximinality.The problem of the bestN-simultaneous approximation in Lp(μ,X) has been deeply and extensively studied; see, for example, [3–11]. When Y is a reflexive subspace of X, it was proved in [3] that Lp(μ,Y) is N–simultaneously proximinal in Lp(μ,X), where μ is the Lebesgue measure on I=[0,1]. We have also obtained similar results, in the Banach space L1(μ0,X) of X-valued Bochner μ0-integrable functions defined on Ω, where μ0 is the restriction of μ to a certain sub-σ-algebra Σ0 of Σ and X is a reflexive space (see [8]).Our purpose is to study theN-simultaneous proximinality of the set Lp(μ,X) defined by (4)Lpμ,W≔g∈Lpμ,X:gs∈Wfor  a.e.s∈Ω,for certain subsets W of X.


## 2. Preliminaries

In this section we wish to include some facts which are fundamental in our study.Definition 1.
Letx1,…,xn be vectors in X; one says that a sequence (yk)k≥1 in Y⊂X is N-simultaneously approximating tox1,…,xn in Y, if (5)limk→∞Nx1-yk,…,xn-yk=infz∈YNx1-z,…,xn-z.Lemma 2 ([8, Lemma 1]).
Letx1,…,xn be vectors in X, and let (yk)k≥1 be a N-simultaneously approximating sequence to x1,…,xn in Y⊂X. Assume that (yk)k≥1 is weakly convergent to y0∈Y. Then y0 is a best N-simultaneous approximation from Y of x1,…,xn.The next result gives a property ofN-simultaneously approximating sequencesin the spaceL1(μ,X). The following lemma is taken from [8, Lemma 2], which was proved for the case when Y=X and Σ0=Σ.Lemma 3 ([8, Lemma 2]).
LetY be a subset of X. Let f1,…,fn be functions in L1(μ,X), and let (gk)k≥1⊂L1(μ,Y) be an N-simultaneously approximating sequence to f1,…,fn in L1(μ,Y). If (Ak)k≥1 is a sequence of Σ-measurable sets such that limk→∞⁡μ(Ak)=0, then (gkχAkc)k≥1 is also an N-simultaneously approximating sequence to f1,…,fn in L1(μ,Y).We also need a result which is probably the best known “subsequence splitting lemma” in integrable function spaces.Lemma 4 (Kadec-Pełczyński-Rosenthal, Lemma2.1.3 in [12]).
If(fm)m≥1 is a bounded sequence in L1(μ), then there exists a subsequence (fmk)k≥1 of (fm)m≥1 and a sequence (Ak)k≥1 of pairwise disjoint measurable sets such that (fmkχAkc)k≥1 is uniformly integrable.


## 3. Main Results

In this section we give the simple and direct proofs of the main results. These results are similar to [6]. However, in [6], more hypotheses are necessary and the proofs are totally different.Theorem 5.
LetW be a weakly compact subset of X. Then L1(μ,W) is N–simultaneously proximinal in L1(μ,X).Proof.
Letf1,…,fn be functions in L1(μ,X), and let (hm)m≥1⊂L1(μ,W) be a N-simultaneously approximating sequence to f1,…,fn in L1(μ,W). We have (6)limm→∞Nf1-hm1,…,fn-hm1=infh∈L1μ,WNf1-h1,…,fn-h1.Since N(f1-hm1,…,fn-hm1) is a convergent sequence, by taking into account the fact that for each m≥1 we have (7)N1,…,1hm1≤Nf1-hm1,…,fn-hm1+Nf11,…,fn1,one deduces that (hm)m≥1 is bounded in L1(μ,W).
By Lemma4, there exists a subsequence (hmk)k≥1 of (hm)m≥1 and a sequence (Ak)k≥1 of pairwise disjoint measurable sets such that (hmkχAkc)k≥1 is uniformly integrable in L1(μ,W).
On the one hand,(8)∑k=1∞μAk=μ⋃k=1∞Ak≤μΩ<∞,so we have limk→∞⁡μ(Ak)=0. Therefore, by Lemma 3, (hmkχAkc)l≥1 is a N-simultaneously approximating sequence to f1,…,fn in L1(μ,W).
Let us denotegk=hmkχAkc for each k≥1.
SinceW⊂X is weakly compact, then the sequence (gk(s))k≥1⊂W for a.e. s∈Ω and has a weakly convergent subsequence, which again is denoted by (gk(s))k≥1. Let us denote by g(s) its weak limit for a.e. s∈Ω. Therefore for each x∗∈X∗ the numerical function x∗(g) is μ-measurable. So g is weakly μ-measurable. On the other hand, for each k∈N, gk is μ-essentially separably valued; that is, there exists Ak∈Σ with μ(Ak)=0 and such that gk(Ω∖Ak) is a norm separable subset of X. For each k let us pick a dense and countable subset, Dk, of gk(Ω∖Ak). Then the set (9)Y=span¯⋃k=1∞Dkis norm closed and separable. For every k∈N and s∈Ω∖Ak we have gk(s)∈Y. Since, g(s) is weak limit of (gk(s))k≥1 for a.e. s∈Ω, we obtain that g(s)∈Y for a.e. s∈Ω. Thus g is μ-essentially separably valued. Therefore, Pettis Measurability Theorem [2, p. 42] guarantees that the function g:Ω→X is μ-measurable.
Since(gk(s))k≥1 is weakly convergent to g(s)∈W for a.e. s∈Ω, then (gk(s))k≥1 is bounded and (10)gs≤liminfk→∞gksfor a.e. s∈Ω. Using this result, the boundedness of (gk)k≥1, and Fatou’s lemma [13, Theorem 2.8.3, p. 131], we get g∈L1(μ,W).
We claim thatg is in the closed convex hull of (gk)k≥1 in (L1(μ,X),·1). Suppose this is not true. Then, by [14, Theorem 9.2, p. 65], there is u∈(L1(μ,X))∗ such that (11)u,g<infk≥1⁡u,gk.
Now, we will use the result that(L1(μ,X))∗ is the space of all bounded continuous mapping u:(Ω,τρ)→(X∗,σ(X∗,X)) (see [1, Theorem 7, p. 94]); we get (12)limk→∞⁡usgks=usgsa.e. on Ω. Using Egoroff’s theorem [13, Theorem 2.2.1, p. 110] and uniform integrability of (gk)k≥1, we get 〈u,gk〉→k→∞〈u,g〉, a contradiction. By taking convex combinations of elements of (gk)k≥1, if necessary, we can assume that gk→k→∞·1g in L1(μ,W). Therefore, by Lemma 2, we have the fact that g is the best N-simultaneous approximation from L1(μ,W) of f1,…,fn.Although the proof of the following theorem is similar to that of Theorem5, we provide the proof here as a means for the reader to readily justify these assertions.Theorem 6.
LetW be a weakly compact subset of X. Then Lp(μ,W) is N–simultaneously proximinal in Lp(μ,X) for each 1<p<∞.Proof.
Letf1,…,fn be functions in Lp(μ,X), and let (gk)k≥1⊂Lp(μ,W) be a N-simultaneously approximating sequence to f1,…,fn in Lp(μ,W). We have (13)limk→∞Nf1-gkp,…,fn-gkp=infh∈Lpμ,WNf1-hp,…,fn-hp.Notice that for each k≥1 we have (14)N1,…,1gkp≤Nf1-gkp,…,fn-gkp+Nf1p,…,fnp.Since N(f1-gkp,…,fn-gkp) is a convergent sequence, we deduce that (gk)k≥1 is bounded in Lp(μ,W). Therefore, (gk)k≥1 is uniformly integrable.
SinceW⊂X is weakly compact, then the sequence (gk(s))k≥1⊂W for a.e. s∈Ω has a weakly convergent subsequence, which again is denoted by (gk(s))k≥1. Let us denote by g(s) its weak limit for a.e. s∈Ω. Therefore for each x∗∈X∗ the numerical function x∗(g) is μ-measurable. So g is weakly μ-measurable. On the other hand, for each k∈N, gk is μ-essentially separably valued; that is, there exists Ak∈Σ with μ(Ak)=0 and such that gk(Ω∖Ak) is a norm separable subset of X. For each k let us pick a dense and countable subset, Dk, of gk(Ω∖Ak). Then the set (15)Y=span¯⋃k=1∞Dkis norm closed and separable. For every k∈N and s∈Ω∖Ak we have gk(s)∈Y. Since g(s) is weak limit of (gk(s))k≥1 for a.e. s∈Ω, we obtain that g(s)∈Y for a.e. s∈Ω. Thus g is μ-essentially separably valued. Therefore, the Pettis Measurability Theorem [2, p. 42] guarantees that the function g:Ω→X is μ-measurable.
Since(gk(s))k≥1 is weakly convergent to g(s)∈W for a.e. s∈Ω, then (gk(s))k≥1 is bounded and (16)gs≤liminfk→∞gksfor a.e. s∈Ω. Using this result, the boundedness of (gk)k≥1, and Fatou’s lemma [13, Theorem 2.8.3, p. 131], we get g∈Lp(μ,W).
We claim thatg is in the closed convex hull of (gk)k≥1 in Lpμ,X,·p. Suppose this is not true. Then, by [14, Theorem 9.2, p. 65], there is a u∈(Lp(μ,X))∗ such that (17)u,g<infk≥1u,gk.
Now, we will be using the result that(Lp(μ,X))∗ is the space of all bounded continuous mapping u:(Ω,τρ)→(X∗,σ(X∗,X)) (see [1, Theorem 9, p. 97]); we get limk→∞⁡u(s)(gk(s))=u(s)(g(s)) a.e. on Ω. Using Egoroff’s theorem [13, Theorem 2.2.1, p. 110] and uniform integrability of (gk)k≥1, we get 〈u,gk〉→k→∞〈u,g〉, a contradiction. By taking convex combinations of elements of (gk)k≥1, if necessary, we can assume that gk→k→∞·pg in Lp(μ,W). Therefore, by Lemma 2, we have the fact that g is the best N-simultaneous approximation from Lp(μ,W) of fp,…,fn.LetΣ0 be a sub–σ–algebra of Σ and μ0 the restriction of μ to Σ0. Let 1≤p<∞. We defined the set (18)Lpμ0,W≔g∈Lpμ0,X:gs∈Wfor  a.e.s∈Ω,where W is a subset of X. We obtain analogously the following theorem.Theorem 7.
LetW be a weakly compact subset of X and 1≤p<∞. Then Lp(μ0,W) is N–simultaneously proximinal in Lp(μ0,X).



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