Linear Spaces, Linear Functionals, and the Hahn–Banach Theorem

Abstract

The Hahn–Banach theorem on the extension of linear functionals that will be proved in the present chapter (alternatively known as the analytic form of the Hahn–Banach theorem) is one of the most important theorems in functional analysis. It is frequently used, both in the subject itself and in applications of functional analysis to a wide circle of related fields. Some of these applications will be treated in this book. The Hahn–Banach theorem is traditionally regarded as one of the “fundamental principles of functional analysis”. Such “fundamental principles” also include the geometric form of the Hahn–Banach theorem (Chapter 9), Banach’s inverse operator theorem, the open mapping and the closed graph theorems, as well as the Banach–Steinhaus theorem (Chapter 10).