Hilbert and Banach Spaces

Abstract

This chapter develops the beginnings of abstract functional analysis, a subject designed to study properties of functions by treating the functions as the members of a space and formulating the properties as properties of the space.

Section 1 defines Banach spaces as complete normed linear spaces and gives anumber of examples of these. The space of bounded linear operators from one normed linear space to another is a normed linear space, and it is a Banach space if the range is a Banach space.

Sections 2–3 concern Hilbert spaces. These are Banach spaces whose norms are induced by inner products. Section 2 shows that closed vector subspaces of such a space have orthogonal complements, and it shows the role of orthonormal bases for such a space. Section 3 concentrates on bounded linear operators from a Hilbert space to itself and constructs the adjoint of each such operator.

Sections 4–6 prove the three main abstract theorems about the norm topology of general normed linear spaces-the Hahn-Banach Theorem, the Uniform Boundedness or Banach-Steinhaus Theorem, and the Interior Mapping Principle. A number of consequences of these theorems are given. The second and third of the theorems require some hypothesis of completeness.