Measures of weak noncompactness in Banach spaces

Abstract

Let X be a real Banach space. A subset B of the dual unit sphere of X is said to be a boundary for X, if every element of X attains its norm on some functional in B. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of X which is compact in the topology of pointwise convergence on B is already weakly compact. This problem was recently solved by Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others.

A quantitative version of Krein’s theorems for Fréchet spaces

“…for any bounded set H in E. Clearly k(H) measures how far H is from being weakly relatively compact in E. The above result from [7,Corollary 3.4] implies that k(coH) ≤ 2k(H) for any bounded set H in a Banach space E, see also [1]. Note that the equality k(coH) = k(H) fails in general, see [10], [11].…”

“…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 Fréchet space E the above function k(H) reads as follows k(H) := sup{d (h, E) : h ∈ H σ(E ,E ) }, where d(h, E) is the natural distance of h to E in the bidual E . The main result of the paper is the following Theorem: For a bounded set H in a Fréchet space E the following inequality holds k(coH) < (2 n+1 − 2)k(H) + 1 2 n for all n ∈ N. Consequently this yields also the following formula k(coH) ≤ k(H)(3 − 2 k(H)).…”

On an integral equation under Henstock–Kurzweil–Pettis integrability

“…For more examples and properties of measures of weak noncompactness, we refer the reader to [2,4,5,21,22]. Definition 2.2 A function f : X 1 −→ X 2 , where X 1 and X 2 are Banach spaces, is said to be weakly-weakly sequentially continuous if for each weakly convergent (x n ) n ⊂ X 1 with x n x, we have f x n f x.…”

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.…”