On weak compactness in L 1 spaces (original) (raw)

On some new characterizations of weakly compact sets in Banach spaces

Studia Mathematica, 2010

In this paper we show several characterizations of weakly compact sets in Banach spaces: Given a bounded closed convex set C of a Banach space X, the following statements are equivalent: i) C is weakly compact; ii) C can be affinely uniformly embedded into a reflexive Banach space; iii) there exists an equivalent norm of X such that it admits the w2R-property on C; iv) there is a continuous and w * -lower semi-continuous (l.s.c.) semi-norm p on the dual X * with p ≥ sup C such that p 2 is everywhere Fréchet differentiable in X * ; and as a consequence, the space X is a weakly compactly generated (WCG) space if and only if there exists a continuous and w * -l.s.c. Fréchet smooth (not necessarily equivalent) norm on X * .

Weak compactness and σ-Asplund generated Banach spaces

Studia Mathematica, 2007

σ-Asplund generated Banach spaces are used to give new characterizations of subspaces of weakly compactly generated spaces and to prove some results on Radon-Nikodým compacta. We show, typically, that in the framework of weakly Lindelöf determined Banach spaces, subspaces of weakly compactly generated spaces are the same as σ-Asplund generated spaces. For this purpose, we study relationships between quantitative versions of Asplund property, dentability, differentiability, and of weak compactness in Banach spaces. As a consequence, we provide a functional-analytic proof of a result of Arvanitakis: A compact space is Eberlein if (and only if) it is simultaneously Corson and quasi-Radon-Nikodým.

Weakly precompact subsets of L1(μ,X)

Colloquium Mathematicum, 2012

Let (Ω, Σ, µ) be a probability space, X a Banach space, and L1(µ, X) the Banach space of Bochner integrable functions f : Ω → X. Let W = {f ∈ L1(µ, X) : for a.e. ω ∈ Ω, f (ω) ≤ 1}. In this paper we characterize the weakly precompact subsets of L1(µ, X). We prove that a bounded subset A of L1(µ, X) is weakly precompact if and only if A is uniformly integrable and for any sequence (fn) in A, there exists a sequence (gn) with gn ∈ co{fi : i ≥ n} for each n such that for a.e. ω ∈ Ω, the sequence (gn(ω)) is weakly Cauchy in X. We also prove that if A is a bounded subset of L1(µ, X), then A is weakly precompact if and only if for every > 0, there exist a positive integer N and a weakly precompact subset H of N W such that A ⊆ H + B(0), where B(0) is the unit ball of L1(µ, X).

Strongly Bounded Representing Measures and Convergence Theorems

Glasgow Mathematical Journal, 2010

Let K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

Weak compactness in the space of operator valued measures Mba(S, L(X,Y)) and its applications

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, 2011

In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures M ba (Σ, L(X, Y)). This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures M ba (Σ, L 1 (X, Y)). This result has interesting applications in optimization and control theory as illustrated by several examples.

Functions and Weakly µH-Compact Spaces

European Journal of Pure and Applied Mathematics, 2017

A GTS (X, µ) is said to be weakly µH-compact if for every µ-open cover {Vα : α ∈ ∆} of X there exists a finite subset ∆0 of ∆ such that X \ ∪{cµ(Vα) : α ∈ ∆0} ∈ H. In this paper we study the effect of functions on weakly µH-compact spaces. The main result is that the θ(µ, ν)-continuous image of a weakly µH-compact space is weakly νf(H)-compact