Banach Space Theory

Abstract

In this chapter we present some basic concepts and results from Functional Analysis (Banach Space Theory). In Sect. 3.2 we introduce the terminology of general locally convex spaces, we define normed and Banach spaces, and then study linear operators between them. We also prove some distinguishing facts about finite dimensional normed spaces. In Sect. 3.3 we present the analytic and geometric forms of the Hahn-Banach theorem. The latter refer to the separation theorems for convex sets. We also prove some of their useful consequences. In Sect. 3.4 we prove the three fundamental theorems of linear functional analysis. These are the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Principle (Banach-Steinhaus Theorem). All of the three are outgrowths of Baire’s Category Theory. In Sect. 3.5 we introduce and discuss the weak and weak* topologies. These are locally convex, nonmetrizable (in infinite dimensions) topologies, which are strictly weaker than the norm (metric) topology and exhibit many interesting features. In Sect. 3.6 we focus on the special classes of separable and reflexive normed spaces. Most of the spaces we encounter in applications are separable and/or reflexive. In Sect. 3.7 we discuss dual operators, compact operators and projection operators. Compact operators exhibit properties similar to those of a finite dimensional operator. In Sect. 3.8 we present the main features of Hilbert spaces. In fact, Hilbert spaces are infinite dimensional generalizations of Euclidean spaces. We also outline the spectral theory for compact self-adjoint operators. Finally, in Sect. 3.9 we have a brief encounter with unbounded linear operators.