Some basic definitions and tools

Abstract

In this short chapter we shall provide some basic information on the theory of locally convex spaces and related topics that will be needed in the sequel. This may unify the approach to some results in Banach spaces, avoiding unnecessary repetitions. For a more complete treatment we advice the reader to look at [Kothe69], [Jarc81], and, for a concise exposition, [RoRo64]. In the second part we shall state the fundamental Krein–Milman theorem on the existence of extreme points of nonempty compact convex subsets of a vector space. This theorem links two quite different objects: extreme points of convex sets and the compactness of the set in some locally convex topology on the space —this is one of the reasons for working in the locally convex context, even if we are interested in Banach spaces—. Finally, we shall collect, for further reference, some important theorems that will appear ubiquitously in the rest of the text. We refer, too, for these and other topics, to [FHHMZ11, Chapter 3].