Non-archimedean Eberlein-Šmulian theory

Abstract

It is shown that, for a large class of non-archimedean normed spaces E, a subset X is weakly compact as soon as f(X) is compact for all f∈E′ (Theorem 2.1), a fact that has no analogue in Functional Analysis over the real or complex numbers. As a Corollary we derive a non-archimedean version of the Eberlein-Šmulian Theorem (2.2 and 2.3, for the ‘classical’ theorem, see [1], VIII, §2 Theorem and Corollary, page 219).