We study the stability of approximative τ-compactness, where τ is the norm or the weak topology. Let Λ be an index set and for every λ∈Λ, let Yλ be a subspace of a Banach space Xλ. For 1≤p<∞, let X=⊕lpXλ and Y=⊕lpYλ. We prove that Y (resp., BY) is approximatively τ-compact in X if and only if, for every λ∈Λ, Yλ (resp., BYλ) is approximatively τ-compact in Xλ.

1. Introduction

Let X be a real Banach space and let K be a subset of X. We denote by τ either the norm or the weak topology on X. The metric projection of X onto K is the set valued map defined by PK(x)={y∈K:x-y=d(x,K)} for x∈X, where d(x,K) denotes the distance from x to K. If, for every x∈X,PK(x)≠∅, we say that K is a proximinal subset of X. A sequence {yn}⊂K is called minimizing for x∈X, if x-yn→d(x,K).

The notion of approximative compactness was introduced by Efimov and Stechkin [1] in connection with the study of Chebyshev sets in Banach spaces and plays an important role in approximation theory (see, e.g., [2, 3]). Deutsch [4] extended this notion to define approximative τ-compactness.

Definition 1.

Let K be a τ-closed subset of X and x0∈X. We say that K is approximatively τ-compact for x0 if every minimizing sequence {yn}⊂K for x0 has a τ-convergent subsequence. If K is approximatively τ-compact for every x∈X, we say that K is approximatively τ-compact in X.

It is easy to verify that approximative τ-compactness implies proximinality. Clearly, compact sets or finite-dimensional subspaces of a Banach space are approximatively compact; weakly compact sets or reflexive subspaces of Banach spaces are approximatively weakly compact. Approximative τ-compactness has been studied in detail in [1, 3–7].

When it comes to the stability of approximative τ-compactness, we suppose that Λ is an index set and for every λ∈Λ,Yλ is a subspace of a Banach space Xλ. And let X=⊕lpXλ,Y=⊕lpYλ, where 1≤p<∞. Bandyopadhyay et al. [5] proved that if Y is approximatively τ-compact in X, then Yλ is approximatively τ-compact in Xλ for every λ∈Λ. In this paper, we prove that the converse is also true. On the other hand, the proximinality of the unit ball of subspaces has been the subject in many recent papers (see, e.g., [8–11]). In this paper, under the above assumption, we also prove that the unit ball of Y is approximatively τ-compact in X if and only if, for every λ∈Λ, the unit ball of Yλ is approximatively τ-compact in Xλ.

For a real Banach space X, we denote by BX the unit ball of X and denote by X∗ the dual space of X. Before we prove the main conclusions we first show a simple property on approximative τ-compactness of the unit ball of subspaces.

Proposition 2.

Let Y be a subspace of a Banach space X. If BY is approximatively τ-compact in X, then so is Y. But the converse is not true.

Proof.

Suppose that x∈X and {yn}⊂Y is a minimizing sequence of x in Y; that is, x-yn→d(x,Y). Then {yn}⊂λBY for sufficiently large λ>0. This means that d(x,Y)=d(x,λBY) and {yn} is also a minimizing sequence of x in λBY. By approximative τ-compactness of λBY (which is equivalent to the one of BY), {yn} has a τ-convergent subsequence.

To illustrate that the converse is not true, first, we show that Bc0 is not approximatively weakly compact in c0. Take x=(2,0,0,…) and for every n∈N, yn=(1,1,1,…,1,0,0,…), where 1 appears n times. Then d(x,Bc0)=1 and {yn} is a minimizing sequence of x in Bc0. But {yn} has no weakly convergent subsequence. Hence Bc0 is not approximatively weakly compact in c0.

Next, let X=c0⊕l1R and Y=c0⊕l1{0}. For any α=(z,r)∈X, it is easy to see that d(α,Y)=r and PY(α)={(z,0)}. Now, suppose {βn}={(zn,0)}⊂Y is a minimizing sequence of α in Y; that is, (1)α-βn=z-zn+r⟶dα,Y=r.This implies that zn→z. Hence βn=(zn,0)→(z,0). Therefore Y is approximatively compact in X. But, by the above discussion, BY is not approximatively weakly compact in Y, and not in X either.

In order to prove our conclusions, we need the following lemmas.

Lemma 3.

Let {Xi:i∈N} be a sequence of Banach spaces and let Yi be a subspace of Xi, respectively, for i∈N. Consider X=⊕lpXi and Y=⊕lpYi, where 1≤p<∞. Let x=(xi)∈X and {yn=(yn,i)}⊂BY be a minimizing sequence of x in BY. Then, for every ε>0, there exists some j∈N such that, for all n, ∑i>jyn,ip<εp.

Proof.

If the conclusion does not hold, then, for every j, there exists infinitely many n such that ∑i>jyn,ip≥εp. We can choose some j0 such that ∑i>j0xip<(ε/3)p and infinite subset {nk} of N such that ∑i>j0ynk,ip≥εp for every k. Therefore for every k,(2)x-ynkp=∑i≤j0xi-ynk,ip+∑i>j0xi-ynk,ip=∑i≤j0xi-ynk,ip+∑i>j0xip+∑i>j0xi-ynk,ip-∑i>j0xip≥dx,BYp+∑i>j0ynk,ip1/p-∑i>j0xip1/pp-∑i>j0xip≥dx,BYp+2ε3p-ε3p≥dx,BYp+ε3p.

But x-yn→d(x,BY)(n→∞); then x-ynp<(d(x,BY))p+(ε/3)p for sufficiently large n. This is a contradiction.

Remark 4.

In Lemma 3, if we replace BY by Y, that is, {yn=(yn,i)}⊂Y is a minimizing sequence of x in Y, then the conclusion still holds.

Lemma 5.

Under the assumption in Lemma 3, if, moreover, limn→∞yn=r and for every i∈N, limn→∞yn,i=ri, then

r=(∑i∈Nrip)1/p;

d(x,BY)=d(x,⊕lpriBYi)=(∑i∈N[d(xi,riBYi)]p)1/p.

Proof.

(1) For every ε>0, by Lemma 3, there exists j∈N such that, for all n, ∑i>jyn,ip<εp. For every fixed j′>j, we can choose n′ such that ∑i≤j′yn′,ip-rip<εp and yn′p-rp<εp. Then (3)∑i≤j′rip-rp≤∑i≤j′rip-yn′p+yn′p-rp=∑i≤j′rip-∑i∈Nyn′,ip+yn′p-rp≤∑i≤j′rip-yn′,ip+∑i>j′yn′,ip+yn′p-rp<3εp.By the arbitrariness of ε, we have r=(∑i∈Nrip)1/p.

(2) Note that r≤1; hence ⊕lpriBYi⊂BY. This implies that d(x,BY)≤d(x,⊕lpriBYi). To prove that d(x,BY)≥d(x,⊕lpriBYi), for every n, we define zn=(zn,i), where zn,i=yn,i for yn,i≤ri, and zn,i=ri/yn,iyn,i for yn,i>ri. Then {zn}⊂⊕lpriBYi.

For arbitrary ε>0, by Lemma 3, there exists j such that ∑i>jrip<εp, and for all n, ∑i>jyn,ip<εp. Further, we can choose some n0 such that, for all n>n0, ∑i≤jyn,i-rip<εp. Then for all n>n0, we have(4)yn-zn=∑i∈Nyn,i-zn,ip1/p≤∑i∈Nyn,i-rip1/p≤∑i≤jyn,i-rip1/p+∑i>jyn,i-rip1/p≤∑i≤jyn,i-rip1/p+∑i>jyn,ip1/p+∑i>jrip1/p<3ε.By the arbitrariness of ε, we have yn-zn→0. This implies that (5)limn→∞x-zn=limn→∞x-yn=dx,BY.Therefore d(x,BY)≥d(x,⊕lpriBYi). So we have d(x,BY)=d(x,⊕lpriBYi).

For the second equality, first, it is obvious that (6)dx,⊕lpriBYi≥∑i∈Ndxi,riBYip1/p.On the other hand, let ε>0 be given. For every i, we can choose zi∈riBYi such that xi-zip<[d(xi,riBYi)]p+ε/2i. Let z=(zi)∈⊕lpriBYi; then (7)x-z=∑i∈Nxi-zip1/p<∑i∈Ndxi,riBYip+ε1/p.By the arbitrariness of ε, we have d(x,⊕lpriBYi)≤(∑i∈N[d(xi,riBYi)]p)1/p. Therefore the second equality holds.

The following is our main result.

Theorem 6.

Let Λ be an index set. For every λ∈Λ, let Yλ be a subspace of a Banach space Xλ. For 1≤p<∞, let X=⊕lpXλ and Y=⊕lpYλ. Then

Y is approximatively τ-compact in X if and only if, for every λ∈Λ, Yλ is approximatively τ-compact in Xλ;

BY is approximatively τ-compact in X if and only if, for every λ∈Λ, BYλ is approximatively τ-compact in Xλ.

Proof.

(1) Necessity has been proven in [5].

Sufficiency: let x∈X and {yn}⊂Y be a minimizing sequence for x. We will show that {yn} has a τ-convergent subsequence. Without loss of generality, we can assume Λ=N and x=(xi),yn=(yn,i).

First, notice that if zi∈PYi(xi) for every i, then z=(zi)∈PY(x). Hence (d(x,Y))p=∑j∈N(d(xj,Yj))p. And for every i, (8)xi-yn,ip+∑j≠idxj,Yjp≤x-ynp=dx,Yp+x-ynp-dx,Yp=∑j∈Ndxj,Yjp+x-ynp-dx,Yp.So (9)xi-yn,ip≤dxi,Yip+x-ynp-dx,Yp⟶dxi,Yipn⟶∞.This implies that, for every i, {yn,i} is a minimizing sequence for xi in Yi. Then {yn,i} has a τ-convergent subsequence by the approximative τ-compactness of Yi. By employing the diagonal process, we can choose a subsequence {ynk} of {yn} such that, for every i, {ynk,i} has a τ-convergent to some yi∈Yi. Obviously, yi∈PYi(xi), and y=(yi)∈Y,x-y=d(x,Y).

We still denote the subsequence {ynk} as {yn}. Next, to complete the proof, we will prove that {yn} has a τ-convergent to y.

Case 1. τ is the norm topology. For every ε>0, by Remark 4, there exists some j such that ∑i>jyip<εp and for all n, ∑i>jyn,ip<εp. Then we can choose n0 such that, for n>n0, ∑i≤jyn,i-yip<εp. Hence for all n>n0, (10)yn-y=∑i≤jyn,i-yip+∑i>jyn,i-yip1/p≤∑i≤jyn,i-yip1/p+∑i>jyn,ip1/p+∑i>jyip1/p<3ε.Therefore, by the arbitrariness of ε, we have that {yn} converges to y.

Case 2. τ is the weak topology. Suppose f=(fi)∈⊕lqXi∗=X∗ with f=1, where 1/p+1/q=1 when p>1 and q=∞ when p=1. For every ε>0, again by Remark 4, we can choose some j such that ∑i>jyip<εp, and for all n, ∑i>jyn,ip<εp. Note that, for every 1≤i≤j, {yn,i} weakly converges to yi; hence there exists n0 such that, for n>n0, ∑i≤jfiyn,i-yi<ε. Then for all n>n0,(11)fyn-y=∑i∈Nfiyn,i-yi≤∑i≤jfiyn,i-yi+∑i>jfiyn,i-yi≤ε+f∑i>jyn,i-yip1/p≤ε+∑i>jyn,ip1/p+∑i>jyip1/p<3ε.

Again by the arbitrariness of ε, we have that {yn} weakly converges to y.

(2) Necessity: fix λ0∈Λ. Suppose that xλ0∈Xλ0, and {yn,λ0}⊂BYλ0 is a minimizing sequence of xλ0 in BYλ0. Let x=(xλ), yn=(yn,λ), where xλ=0 and yn,λ=0 for λ≠λ0. Then x∈X,{yn}⊂BY, and (12)x-yn=xλ0-yn,λ0⟶dxλ0,BYλ0.Notice that d(x,BY)≥d(xλ0,BYλ0). Hence (13)x-yn⟶dxλ0,BYλ0=dx,BY,which implies that {yn} is a minimizing sequence of x in BY. By approximative τ-compactness of BY in X, {yn} has a τ-convergent subsequence {ynk}. Therefore {ynk,λ0} is τ-convergent.

Sufficiency: suppose that x∈X∖BY and {yn}⊂BY is a minimizing sequence of x in BY. Like the proof in (1), we will prove that {yn} has a τ-convergent subsequence and we can assume Λ=N, x=(xi), and yn=(yn,i). By employing the diagonal process, we can choose a subsequence {ynk} of {yn} (we still denote the subsequence as {yn}) such that limn→∞yn=r, and for every i∈N, limn→∞yn,i=ri. Then by Lemma 5, we have r=(∑i∈Nrip)1/p, and(14)dx,BY=dx,⊕lpriBYi=∑i∈Ndxi,riBYip1/p.

Next, for every i∈N, we will show that {yn,i} has a τ-convergent subsequence. We can assume that, for all n and i, yn,i≤ri. Otherwise, we can replace {yn} with {zn} which we define in the proof of Lemma 5(2).

Note that, for every i∈N, (15)xi-yn,ip+∑j≠idxj,rjBYjp≤x-ynp=dx,BYp+x-ynp-dx,BYp=∑j∈Ndxj,rjBYjp+x-ynp-dx,BYp.Then (16)dxi,riBYip≤xi-yn,ip≤dxi,riBYip+x-ynp-dx,BYp.Hence, when n→∞, (17)xi-yn,i⟶dxi,riBYi.This implies that {yn,i} is a minimizing sequence of xi in riBYi. By approximative τ-compactness of riBYi in Xi, {yn,i} has a τ-convergent subsequence.

Employing the diagonal process again, we can choose a subsequence {ynk} of {yn} such that, for evrey i, {ynk,i} has a τ-convergents to some yi∈riBYi. Let y=(yi); then y∈⊕lpriBYi. We still denote {ynk} as {yn}. Finally, just like the proof in (1), we can prove that {yn} has a τ-convergent to y.

Remark 7.

The above theorem does not hold for p=∞. Indeed, suppose that Z is an infinite-dimensional proper closed subspace of l2. By Theorem 1.4 in [12], BZ is approximatively compact in l2. Next, we show that Z⊕l∞Z is not approximatively compact in l2⊕l∞l2. Choose x0∈l2∖Z and y0∈Z such that d(x0,Z)=1=x0-y0. Furthermore, we take a sequence {zn}⊂Z with zn=1 satisfying that {zn} has no convergent subsequence. Note that (18)0,x0-zn,y0∞=maxzn,x0-y0=1;and for any (z,y)∈Z⊕l∞Z, (19)0,x0-z,y∞=maxz,x0-y≥x0-y≥1.This means that d((0,x0),Z⊕l∞Z)=1 and {(zn,y0)} is a minimizing sequence of (0,x0) in Z⊕l∞Z. But {(zn,y0)} has no convergent subsequence.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank colleagues and graduate students in the Functional Analysis group of Xiamen University for their very helpful conversations and suggestions. Zhenghua Luo was supported partially by the Natural Science Foundation of China, Grant no. 11201160, and the Natural Science Foundation of Fujian Province, Grant no. 2012J05006. Wen Zhang was supported in part by the Natural Science Foundation of China, Grant no. 11471270, and by the Natural Science Foundation of Fujian Province, Grant no. 2015J01022.

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