Characterizing weak compactness

Dependent Choices and Weak Compactness

We work in set-theory without the Axiom of Choice ZF. We prove that the principle of Dependent Choices (DC) implies that the closed unit ball of a uniformly convex Banach space is weakly compact, and in particular, that the closed unit ball of a Hilbert space is weakly compact. These statements are not provable in ZF, and the latter statement does not imply DC. Furthermore, DC does not imply th...

A Weak Grothendieck Compactness Principle

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 this article, an analogue of the Grothendieck compactness principle is considered when the norm topology of a Banach space is replaced by its weak topology. It is shown that every weakly compact subset of a Banach space is contained in...

Smoothness and Weak* Sequential Compactness

If a Banach space E has an equivalent smooth norm, then every bounded sequence in E* has a weak* converging subsequence. Generalizations of this result are obtained.

Weak Compactness of the Set of ε-Extensions

Characterizing Continuity by Preserving Compactness and Connectedness

Let us call a function f from a space X into a space Y preserving if the image of every compact subspace of X is compact in Y and the image of every connected subspace of X is connected in Y . By elementary theorems a continuous function is always preserving. Evelyn R. McMillan [6] proved in 1970 that if X is Hausdorff, locally connected and Frèchet, Y is Hausdorff, then the converse is also tr...