Functional analysis

Belton A.

These notes are an expanded version of a set written for a course given to final-yearundergraduates at the University of Oxford.A thorough understanding of the Oxford third-year b4 analysis course (an introductionto Banach and Hilbert spaces) or its equivalent is a prerequisite for this material. Weuse [24] as a compendium of results from that series of lectures. (Numbers in squarebrackets refer to items in the bibliography.)The author acknowledges his debt to all those from whom he has learnt functionalanalysis, especially Professor D. A. Edwards, Dr G. R. Allen and Dr J. M. Lindsay. Thestudents attending the course were very helpful, especially Mr A. Evseev, Mr L. Taitzand Ms P. Iley.This document was typeset using LATEX2ε with Peter Wilson’s memoir class and theAMS-LATEX and XY-pic packages. The index was produced with the aid of the MakeIndexprogram.This edition contains a few additional exercises and the electronic version is equippedwith hyperlinks, thanks to the hyperref package of Sebastian Rahtz and Heiko Oberdiek.

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Table of contents :
Contents……Page 3
Introduction……Page 5
Basic Definitions……Page 9
Subspaces and Quotient Spaces……Page 10
Completions……Page 12
Direct Sums……Page 13
Initial Topologies……Page 15
Nets……Page 16
Exercises 1……Page 20
Preliminaries……Page 23
Extension of Linear Operators……Page 24
The Baire Category Theorem……Page 25
The Open-Mapping Theorem……Page 27
Exercises 2……Page 28
The Closed-Graph Theorem……Page 32
The Strong Operator Topology……Page 33
Exercises 3……Page 34
The Weak Topology……Page 37
The Hahn-Banach Theorem……Page 38
Adjoint Operators……Page 42
Exercises 4……Page 43
Tychonov’s Theorem……Page 45
The Banach-Alaoglu Theorem……Page 46
Topological Vector Spaces……Page 47
The Krein-Milman Theorem……Page 51
Exercises 5……Page 53
Normed Algebras……Page 59
Quotient Algebras……Page 60
Unitization……Page 61
Approximate Identities……Page 62
Completion……Page 63
Invertibility……Page 65
The Spectrum and Resolvent……Page 66
The Spectral-Radius Formula……Page 67
Exercises 6……Page 69
Characters and Maximal Ideals……Page 71
Characters and the Spectrum……Page 72
The Gelfand Topology……Page 73
Examples……Page 74
Exercises 7……Page 78
Tychonov via Nets……Page 81
Exercises A……Page 84
Exercises 1……Page 85
Exercises 2……Page 89
Exercises 3……Page 94
Exercises 4……Page 98
Exercises 5……Page 103
Exercises 6……Page 111
Exercises 7……Page 115
Exercises A……Page 122
Bibliography……Page 123
Index……Page 125