Weak compactness in Banach spaces

Compactness in Vector-valued Banach Function Spaces

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces L X , where X is a Banach space and 1 ≤ p < ∞, and extend the result to vector-valued Banach function spaces EX , where E is a Banach function space with order continuous norm. Let X be a Banach space. The problem of describing the compact sets in the Lebesgue-Bochner spaces LpX , ...

Weak Interpolation in Banach Spaces

On weak compactness in L1 spaces

We will use the concept of strong generating and a simple renorming theorem to give new proofs to slight generalizations of some results of Argyros and Rosenthal on weakly compact sets in L1(μ) spaces for finite measures μ. The purpose of this note is to show that a simple transfer renorming theorem explains why L1(μ)-spaces, for finite measures μ, share some properties with superreflexive spac...

Weak∗ Smooth Compactness in Smooth Topological Spaces

In this paper we obtain some properties of the weak smooth α-closure and weak smooth α-interior of a fuzzy set in smooth topological spaces and introduce the concepts of several types of weak∗ smooth compactness in smooth topological spaces and investigate some of their properties.

A Weak Grothendieck Compactness Principle for Banach spaces with a Symmetric Basis

The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In [1], an analogue of the Grothendieck compactness principle for the weak topology was used to characterize Banach spaces with the Schur property. Using a different analogue of the Grothendieck compactness principle for the weak topology...