C-Infinity-Bounding Sets and Compactness

Abstract

In this article it is proved that any set in a real Banach space on which the C-infinity-functions are bounded, is relatively compact. In particular, for any real Banach space E, a sequence (x(n)) convergences to x in E if and only if f(x(n)) --> S(x) for all f is an element of C-infinity(E).