Notions from Topology and Functional Analysis

Abstract

In this chapter, we furnish the basis needed to understand the themes studied in this book. In addition to giving the definitions of topological vector spaces, normed spaces, and Banach spaces and their topological duals, we present fundamental theorems from functional analysis. We introduce the notion of reflexivity and recall results on Hilbert spaces and uniformly convex spaces, whose properties we use in many examples in this book. We also recall results on distribution spaces, which we need in the definition of Sobolev spaces. We conclude with a section on the spaces L ^p and their reflexivity. In particular, we give a compactness criterion for bounded subsets of L ^p, which becomes a crucial argument when we study the solutions of the variational problems associated with elliptic partial differential equations (Chapter 5).

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4471-2807-6_8