summary:We give a characterization of $K$-weakly precompact sets in terms of uniform Gateaux differentiability of certain continuous convex functions

Matsuda, Minoru

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeMinoru MatsudaA remark on localized weak precompactness in Banach spacesCommentationes Mathematicae Universitatis Carolinae, Vol. 40 (1999), No. 2, 271--276Persistent URL: http://dml.cz/dmlcz/119082Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1999Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 40,2 (1999)271–276 271A remark on localized weakprecompactness in Banach spacesMinoru MatsudaAbstract. We give a characterization of K-weakly precompact sets in terms of uniformGateaux differentiability of certain continuous convex functions.Keywords: K-weakly precompact set, uniform Gateaux differentiabilityClassification: 46B07, 46B22, 49J50We begin with the requisite definition. Throughout this paper X denotes areal Banach space with topological dual X∗. If g : X → R is a continuous convexfunction, for x, y ∈ X , we define Dg (x, y) bylimt→0{g(x+ ty)− g(x)}/tprovided that this limit exists, and we also define the subdifferential of g at x(∈ X) to be the set ∂g(x) of all elements x∗ of X∗ satisfying that (u, x∗) ≤ g(x+u)−g(x) for any u ∈ X . Then ∂g(x) is a non-empty weak∗-compact convex subsetof X∗ for every x ∈ X . The triple (I,Λ, λ) refers to the Lebesgue measure spaceon I (= [0, 1]), Λ+ to the sets in Λ with positive λ-measure. We always understandthat I is endowed with Λ and λ. We denote the set {χE/λ(E) : E ∈ Λ+} by ∆(I).A function f : I → X∗ is said to be weak∗-measurable if (x, f(t)) is λ-measurablefor each x ∈ X . If f : I → X∗ is a bounded weak∗-measurable function, we obtaina bounded linear operator Tf : X → L1(I,Λ, λ) given by Tf (x) = x ◦ f for everyx ∈ X , where (x ◦ f)(t) = (x, f(t)) for every t ∈ I, and the dual operator of Tf isdenoted by T ∗f (: L∞(I,Λ, λ)→ X∗).According to Bator and Lewis [1], let us define the notion of localized weakprecompactness in Banach spaces as follows.Definition 1. Let A be a bounded subset of X and K a weak∗-compact subsetof X . Then we say that A is K-weakly precompact if every sequence {xn}n≥1 inA has a pointwise convergent subsequence {xn(k)}k≥1 on K.Then, in [1], they have made a systematic study of K-weakly precompact setsA in Banach spaces and obtained various characterizations of such sets.Succeedingly, in our paper [4], we also have obtained measure theoretic cha-racterizations of K-weakly precompact sets A by the effective use of a K-valued272 M.Matsudaweak∗-measurable function constructed in the case where A is non-K-weakly pre-compact. In this paper we wish to add a characterization ofK-weakly precompactsets in terms of uniform Gateaux differentiability of certain continuous convexfunctions, which is our aim. This can be regarded as a slight generalization andrefinement of Corollary 10 in [1]. And it should be noted that even here thisK-valued function also becomes an effective means to an end. Before giving ourcharacterization theorem, let us define some special continuous convex functionson X as follows.Definition 2. Let H be a non-empty bounded subset of X∗. Then the contin-uous convex function associated with H , which is denoted by gH , is defined bygH(x) = sup{(x, x∗) : x∗ ∈ H} for every x ∈ X .In what follows, all notations and terminology used and not defined are asin [1].Let A be a bounded subset of X , K a weak∗-compact subset of X∗, {xn}n≥1 asequence in A and Y the closed linear span of {xn : n ≥ 1} in X . In the following,we always understand that Y is a such space. Let j : Y → X be the inclusion map-ping and j∗ its dual mapping. For any non-empty subset H of K, the continuousconvex function gH : Y → R satisfies that ∂gH(y) ⊂ co∗(j∗(K)) for each y ∈ Y .Further let us note two preliminary facts for the proof of Theorem. One concernsseparably related sets in the case where A is K-weakly precompact. Let {xn}n≥1be a sequence in A and suppose that there exists a subsequence {xn(k)}k≥1 of{xn}n≥1 such that limk→∞(xn(k), x∗) exists for every x∗ ∈ K. Then this im-plies that limk→∞(xn(k), y∗) exists for every y∗ ∈ co∗(j∗(K)). Hence, by con-sidering the mapping L : co∗(j∗(K)) → c (the Banach space of all convergentsequences of real numbers equipped with the supremum norm ‖ · ‖∞) defined byL(y∗) = {(xn(k), y∗)}k≥1, we easily know that co∗(j∗(K)) is separably related to{xn(k) : k ≥ 1}, since c is separable. The other concerns the construction of aK-valued weak∗-measurable function h and a sequence {xn}n≥1 in A in the casewhere A is non-K-weakly precompact. Then, although the construction of thisfunction h and the sequence {xn}n≥1 in A is exactly the same as in § 3 of [4], forthe sake of completeness, we state its outline briefly in the following. Since A isnot K-weakly precompact, by the celebrated argument of Rosenthal [5], we havea sequence {xn}n≥1 in A and real numbers r and δ with δ > 0 such that puttingAn = {x∗ ∈ K : (xn, x∗) ≤ r} andBn = {x∗ ∈ K : (xn, x∗) ≥ r+δ}, (An, Bn)n≥1is an independent sequence of pairs of weak∗-closed subsets ofK (that is, for every{εj}1≤j≤k with εj = 1 or −1,⋂{εjAj : 1 ≤ j ≤ k} is a non-empty set, whereεjAj = Aj if εj = 1 and εjAj = Bj if εj = −1). Putting Γ =⋂n≥1(An ∪ Bn),Γ is a non-empty weak∗-compact subset of K, since (An, Bn)n≥1 is indepen-dent. Define ϕ : Γ → P(N) (Cantor space, with its usual compact metric topol-ogy) by ϕ(x∗) = {p : (xp, x∗) ≤ r} (= {p : Ap ∋ x∗}) ∈ P(N). Then ϕ isa continuous surjection from Γ to P(N) (here, Γ is endowed with the weak∗-topology σ(X∗, X)) and so we have a Radon probability measure γ on Γ suchthat ϕ(γ) = ν (the normalized Haar measure if we identify P(N) with {0, 1}N)A remark on localized weak precompactness in Banach spaces 273and {f ◦ ϕ : f ∈ L1(P(N),Σν , ν)} = L1(Γ,Σγ , γ) where Σν (resp. Σγ) is thefamily of all ν (resp. γ)-measurable subsets of P(N) (resp. Γ). Further, con-sider a function τ : P(N) → I defined by τ(D) = Σ{1/2m : m ∈ D} forevery D ∈ P(N). Then τ is a continuous surjection such that τ(ν) = λ and{u ◦ τ : u ∈ L1(I,Λ, λ)} = L1(P(N),Σν , ν). Then, making use of the liftingtheory, we have a weak∗-measurable function h : I → Γ (⊂ K) such thatρ(x ◦ h)(t) = (x, h(t)) for every x ∈ X and every t ∈ I,(α)∫E(x, h(t)) dλ(t) =∫ϕ−1(τ−1(E))(x, x∗) dγ(x∗)(β)for every E ∈ Λ and every x ∈ X . Here ρ denotes a lifting on L∞(I,Λ, λ).Now we are ready to state our characterization theorem (a localized version ofTheorem 8 in [1]). Its main part is that (3) implies (1), whose proof is significantin the point that the characters of the K-valued function h and the sequence{xn}n≥1 in A obtained above are used concretely and effectively. And there,we can get a result that for every y ∈ Y and every subsequence {xn(k)}k≥1 of{xn}n≥1, DgH (y, xn(k)) does not exist uniformly in k, where H = h(I) (⊂ K).Theorem. Let A be a bounded subset of X and K a weak∗-compact (not ne-cessarily convex) subset of X∗. Then the following statements about A and Kare equivalent.(1) The set A is K-weakly precompact.(2) If {xn}n≥1 is a sequence in A and g : Y → R is a continuous convex functionsuch that ∂g(y) ⊂ co∗(j∗(K)) for every y ∈ Y , then there exists a dense Gδ-subsetG of Y and a subsequence {xn(k)}k≥1 of {xn}n≥1 such that Dg (y, xn(k)) existsuniformly in k for each y ∈ G.(3) If {xn}n≥1 is a sequence in A and H is a non-empty subset of K, thenthere exists an element y of Y and a subsequence {xn(k)}k≥1 of {xn}n≥1 suchthat DgH(y, xn(k)) exists uniformly in k.Proof: (1) ⇒ (2). The proof is analogous to that of the corresponding partof Theorem 8 in [1]. Suppose that (1) holds. Take any sequence {xn}n≥1 in Aand any continuous convex function g : Y → R such that ∂g(y) ⊂ co∗(j∗(K)) forevery y ∈ Y . As A is K-weakly precompact, we have a subsequence {xn(k)}k≥1 of{xn}n≥1 such that limk→∞(xn(k), x∗) exists for every x∗ ∈ K. Therefore, by thefirst preliminary fact preceding Theorem, co∗(j∗(K)) is separably related to B(= {xn(k) : k ≥ 1}). So it is separably related to aco(B) (: the absolutely convexhull of B). Since ∂g(y) ⊂ co∗(j∗(K)) for every y ∈ Y , by the same argument asin Theorem 3.14 and Proposition 3.15 of [2], we have a dense Gδ-subset G of Ysuch that g is aco(B)-differentiable (cf. [2]) at every y ∈ G, whence (2) holds.(2) ⇒ (3). This follows immediately from the fact that ∂gH(y) ⊂ co∗(j∗(K))for every non-empty subset H of K and every y ∈ Y .274 M.Matsuda(3) ⇒ (1). The proof of this part is crucial. Suppose that (1) fails. By thesecond preliminary fact preceding Theorem, we have a function h : I → K anda sequence {xn}n≥1 in A as stated above. Take H = h(I), and let {U(n, k) :n = 0, 1, . . . ; k = 0, . . . , 2n − 1} be a system of open intervals in I given byU(n, k) = (k/2n, (k + 1)/2n) if n ≥ 0, 0 ≤ k ≤ 2n − 1. Then we get thatϕ−1(τ−1(U(n, 2k))) ⊂ Bn and ϕ−1(τ−1(U(n, 2k+1))) ⊂ An for n = 1, 2, . . . andk = 0, . . . , 2n−1 − 1. Further we note a following elementary fact: Let E ∈ Λ+and {n(i)}i≥1 be a strictly increasing sequence of natural numbers. Then thereexists a natural number i and a non-negative number q with 0 ≤ 2q < 2n(i) − 1such that both E ∩ U(n(i), 2q) and E ∩ U(n(i), 2q + 1) are in Λ+, which can beeasily shown by an argument used in Lemma 2 of [3].Now, let us show that for every subsequence {xn(k)}k≥1 of {xn}n≥1 and everyy ∈ Y , DgH (y, xn(k)) does not exist uniformly in k. To this end, take any pointy in Y and any subsequence {xn(k)}k≥1 of {xn}n≥1 , and set yk = xn(k) forevery k. Consider a family of weak∗-open slices of co∗(j∗(T ∗h (∆(I)))) (= M) :{S(y, δ/3i, M) : i ≥ 1}. Then we have that for every iS(y, δ/3i, M) ={y∗ ∈ M : (y, y∗) > supz∗∈M(y, z∗)− δ/3i}={y∗ ∈ M : (y, y∗) > ess-supt∈I(j(y), h(t)) − δ/3i}={y∗ ∈ M : (y, y∗) > gH(y)− δ/3i},since gH(y) = supt∈I (j(y), h(t)) = ess-supt∈I (j(y), h(t)) by virtue of (α) above.So, letting Ei = {t ∈ I : (j(y), h(t)) > gH(y) − δ/3i}, we easily get that Ei ∈Λ+ and j∗(h(Ei)) ⊂ S(y, δ/3i, M) for every i. Hence, by the elementary factstated above, there exists a natural number k(i) and a non-negative number q(i)with 0 ≤ 2q(i) < 2n(k(i)) − 1 such that both Ei ∩ U(n(k(i)), 2q(i)) and Ei ∩U(n(k(i)), 2q(i) + 1) are in Λ+. For every i, let Fi = Ei ∩ U(n(k(i)), 2q(i))and Gi = Ei ∩ U(n(k(i)), 2q(i) + 1), and let u∗i = j∗(T ∗h (χFi/λ(Fi))) and v∗i =j∗(T ∗h(χGi/λ(Gi))). Then we have that for every i(a) (y, u∗i ) > gH(y)− δ/3i and (y, v∗i ) > gH(y)− δ/3i,(b) (yk(i), u∗i − v∗i ) ≥ δ,(c) gH(y + yk(i)/i) ≥ (y + yk(i)/i, u∗i ) and gH(y − yk(i)/i) ≥ (y − yk(i)/i, v∗i ).Indeed, we have that(y, u∗i ) = (j(y), T∗h (χFi/λ(Fi)))={∫Fi(j(y), h(t)) dλ(t)}/λ(Fi) > gH(y)− δ/3i,since j∗(h(Fi)) ⊂ S(y, δ/3i, M). Similarly, (y, v∗i ) > gH(y) − δ/3i. Thus wehave (a). And we can prove (b) as follows. In virtue of (β), we have that forA remark on localized weak precompactness in Banach spaces 275every i(yk(i), u∗i − v∗i )= (j(yk(i)), T∗h (χFi/λ(Fi)))− (j(yk(i)), T∗h (χGi/λ(Gi)))= (j(xn(k(i))), T∗h (χFi/λ(Fi))) − (j(xn(k(i))), T∗h (χGi/λ(Gi)))={∫Fi(j(xn(k(i))), h(t)) dλ(t)}/λ(Fi)−{∫Gi(j(xn(k(i))), h(t)) dλ(t)}/λ(Gi)={∫ϕ−1(τ−1(Fi))(j(xn(k(i))), x∗) dγ(x∗)}/λ(Fi)−{∫ϕ−1(τ−1(Gi))(j(xn(k(i))), x∗) dγ(x∗)}/λ(Gi)≥ (r + δ)− r = δ,since ϕ−1(τ−1(Fi)) (⊂ ϕ−1(τ−1(U(n(k(i)), 2q(i))))) ⊂ Bn(k(i)), ϕ−1(τ−1(Gi))(⊂ ϕ−1(τ−1(U(n(k(i)), 2q(i) + 1)))) ⊂ An(k(i)) and τ(ϕ(γ)) = λ. As to (c), wehave that for every igH(y + yk(i)/i) = supt∈I(j(y + yk(i)/i), h(t))≥{∫Fi(j(y + yk(i)/i), h(t)) dλ(t)}/λ(Fi) = (y + yk(i)/i, u∗i ).Similarly, gH(y − yk(i)/i) ≥ (y − yk(i)/i, v∗i ). Then, making use of (a), (b) and(c), we have that for every igH(y + yk(i)/i) + gH(y − yk(i)/i)− 2 · gH(y)> (y + yk(i)/i, u∗i ) + (y − yk(i)/i, v∗i )− {(y, u∗i + v∗i ) + 2δ/3i}= (yk(i), u∗i − v∗i )/i − 2δ/3i ≥ δ/3i.Consequently, we have that for every i{gH(y + yk(i)/i) + gH(y − yk(i)/i)− 2 · gH(y)}/(1/i) > δ/3,which implies that DgH(y, xn(k)) does not exist uniformly in k. Thus the proofis complete. Acknowledgment. The author is grateful to the referee for suggestions in re-vising the paper.276 M.MatsudaReferences[1] Bator E.M., Lewis P.W., Weak precompactness and the weak RNP, Bull. Polish Acad. Sci.Math. 37 (1989), 443–452.[2] Fitzpatrick S.P., Separably related sets and the Radon-Nikodym property, Illinois J. Math.29 (1985), 229–247.[3] Matsuda M., A characterization of Pettis sets in terms of the Bourgain property, Math.Japon. 41 (1995), 433–439.[4] Matsuda M., On localized weak precompactness in Banach spaces, Publ. RIMS, KyotoUniv. 32 (1996), 473–491.[5] Rosenthal H., A characterization of Banach spaces containing l1, Proc. Nat. Acad. Sci. 71(1974), 2411–2413.Department of Mathematics, Faculty of Science, Shizuoka University, Ohya,Shizuoka 422–8529, JapanE-mail : smmmatu@ipcs.shizuoka.ac.jp(Received January 21, 1998, revised May 19, 1998)

A remark on localized weak precompactness in Banach spaces

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A remark on localized weak precompactness in Banach spaces

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Institute of Mathematics AS CR, v. v. i.