In a Banach space, the closed convex hull of weakly compact set is weakly compact

Question

I am wondering if the following statement is true

Let $X$ be a weakly compact subset of a Banach space $E$ (not necessarily reflexive), then the closed convex hull of $X$ is a weakly compact subset of $E$ ?

Any answer or literature related to that would be most welcome.

Answer 1

Yes. The result is known as the Krein-Šmulian Theorem. A proof can be found in Dunford and Schwartz' Linear Operators Part I (V 6.4). An outline of this proof (I believe) is given in exercise IV.3 of Diestels' Sequences and Series in Banach Spaces.

According to Dunford and Schwartz, the result was first shown in the separable case by Krein (Sur quelques questions de la géométrie des ensembles convexes situés dans un espace linéaire normé et complet. Doklady Akad. Nauk SSSR (N.S.) 14, 5-7 (1937)) and in the general case by Krein and Šmulian (On Regularly convex sets in the space conjugate to a Banach Space). Ann. of Math. 41, 556-583 (1940)).