An extension of the Krein-Šmulian Theorem

Abstract

Let X be a Banach space, u ∈ X * * and K, Z two subsets of X * *. Denote by d(u, Z) and d(K, Z) the distances to Z from the point u and from the subset K respectively. The Krein-Šmulian Theorem asserts that the closed convex hull of a weakly compact subset of a Banach space is weakly compact; in other words, every w *-compact subset K ⊂ X * * such that d(K, X) = 0 satisfies d(co w * (K), X) = 0. We extend this result in the following way: if Z ⊂ X is a closed subspace of X and K ⊂ X * * is a w * −compact subset of X * * , then d(co w * (K), Z) ≤ 5d(K, Z). Moreover, if Z ∩ K is w *-dense in K, then d(co w * (K), Z) ≤ 2d(K, Z). However, the equality d(K, X) = d(co w * (K), X) holds in many cases, for instance, if 1 ⊆ X * , if X has w *-angelic dual unit ball (for example, if X is WCG or WLD), if X = 1 (I), if K is fragmented by the norm of X * * , etc. We also construct under CH a w *-compact subset K ⊂ B(X * *) such that K ∩ X is w *-dense in K, d(K, X) = 1 2 and d(co w * (K), X) = 1.

A quantitative approach to weak compactness in Fréchet spaces and spacesC(X)

“…This line of research motivates a number of specialists to study several quantitative counterparts of some classical results. We refer to works [9], [2], [3], [4], [11], [15], [16] also as a good source of references. Especially results from [9] and [2], yielding several characterizations of weak compactness of bounded sets in a Banach space, motivated our present paper.…”

The distance $dist (\mathcal{B},X)$ when $\mathcal{B}$ is a boundary of $B(X^{**})$

“…When K is a w * -compact subset of a bidual Banach space X * * and B = K, the distance dist(K, X) is, in general, different from the distance dist(co w * (K), X). Actually, it is known (see [2,5]) that [7,1]). The underlying scalar field we consider is the reals.…”

“…The topic of our theorem is part of the general question to which extent and in which sense co w * (B) can be recovered by a boundary B (cf. [2,3,4,5,6,9]). When K is a w * -compact subset of a bidual Banach space X * * and B = K, the distance dist(K, X) is, in general, different from the distance dist(co w * (K), X).…”

A quantitative version of Krein’s theorems for Fréchet spaces

“…In the last decade several quantitative counterparts of some other classical results (including Gantmacher, Eberlein-Grothendieck, Grothendieck, KreinSmulyan' theorems) have been proved by several specialists, see for example [1], [2], [3], [4], [5], [8], [10], [11] and references. It turns out that these new versions strengthen the original results and provide new applications both in functional analysis and topology.…”

“…For a Banach space E and its bidual space E the following function k(H) := sup y∈H σ(E ,E ) infx∈E y−x defined on bounded subsets H of E measures how far H is from being σ(E, E )-relatively compact in E. This concept, introduced independently by Granero (2006) and Cascales-Marciszewski-Raja (2006), has been used to study a quantitative version of Krein's theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper a quantitative version of Krein's theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces.…”