PROOF THEORY OF WEAK COMPACTNESS

Weak Compactness of the Set of ε-Extensions

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.

On the Strength of Weak Compactness

We study the logical and computational strength of weak compactness in the separable Hilbert space `2. Let weak-BW be the statement the every bounded sequence in `2 has a weak cluster point. It is known that weak-BW is equivalent to ACA0 over RCA0 and thus that it is equivalent to (nested uses of) the usual Bolzano-Weierstraß principle BW. We show that weak-BW is instance-wise equivalent to Π2-...