Principles of Banach Spaces

Abstract

The theme of our presentation up to this point may be described as a study of the interplay between the algebraic-geometric notions of convex sets and mappings, extreme points, etc. and the topological notions of openness, compactness, continuity, etc. For such a study the correct setting is, as we have seen, the linear topological space (frequently required also to be locally convex). The resulting theory is broad and powerful, as we hope has been demonstrated by Chapter II. Further development now requires some additional specialization of our setting. The crucial new hypothesis which we now bring in is that of completeness. We shall also generally limit our considerations to normed linear spaces, unless the results under consideration can be clearly and cleanly extended to locally convex spaces. More typically, locally convex topologies will play a vital supporting role in our theory of Banach spaces, particularly the weak and weak* topologies.