Functional analytic basis
Johann Boos, Peter Cass · 2000
Abstract The aim of this chapter is to familiarize the reader with functional analytic tools which we will apply in the remaining parts of the book. We assume that the reader is already familiar with basic notions about topological spaces, semi-metric spaces and semi-normed spaces. However, to make the book more self-contained, we give, in Sections 6.1-6.3, a compact introduction to these concepts. We concentrate on basic notions, facts, examples and theorems and omit almost all proofs. In view of our main concern, most of the examples are related to sequence spaces. Among other things we recall, in Section 6.2, the Baire category theorem for metric spaces and, in Section 6.3, the Hahn-Banach theorem, the uniform boundedness principle and the Banach-Steinhaus theorem for semi-normed spaces.