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Chapter XII. Hilbert and Banach Spaces

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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 a number 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.

The topic of Hilbert and Banach spaces continues in Chapter IV of the companion volume, Advanced Real Analysis.

Chapter information

Source
Anthony W. Knapp, Basic Real Analysis, Digital Second Edition (East Setauket, NY: Anthony W. Knapp, 2016), 570-602

Dates
First available in Project Euclid: 26 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.bia/1532570522

Digital Object Identifier
doi:10.3792/euclid/9781429799997-12

Rights
Copyright © 2016, Anthony W. Knapp

Citation

Knapp, Anthony W. Chapter XII. Hilbert and Banach Spaces. Basic Real Analysis, 570--602, Anthony W. Knapp, East Setauket, New York, 2016. doi:10.3792/euclid/9781429799997-12. https://projecteuclid.org/euclid.bia/1532570522


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