A Course in Functional Analysis

Front Cover
Springer Science & Business Media, Jan 25, 1994 - Mathematics - 400 pages
1 Review
Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both. In this book I have tried to follow the common thread rather than any special topic. I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic. Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Hilbert Spaces
xv
2 Orthogonality
5
3 The Riesz Representation Theorem
9
4 Orthonormal Sets of Vectors and Bases
12
5 Isomorphic Hilbert Spaces and the Fourier Transform for the Circle
17
6 The Direct Sum of Hilbert Spaces
21
Operators on Hilbert Space
24
2 The Adjoint of an Operator
29
3 Compact Operators
171
4 Invariant Subspaces
176
5 Weakly Compact Operators
181
Banach Algebras and Spectral Theory for Operators on a Banach Space
185
2 Ideals and Quotients
189
3 The Spectrum
193
4 The Riesz Functional Calculus
197
5 Dependence of the Spectrum on the Algebra
203

3 Projections and Idempotents Invariant and Reducing Subspaces
34
4 Compact Operators
39
5 The Diagonalization of Compact SelfAdjoint Operators
44
SturmLiouville Systems
47
7 The Spectral Theorem and Functional Calculus for Compact Normal Operators
52
8 Unitary Equivalence for Compact Normal Operators
58
Banach Spaces
61
2 Linear Operators on Normed Spaces
65
3 Finite Dimensional Normed Spaces
67
4 Quotients and Products of Normed Spaces
68
5 Linear Functionals
71
6 The HahnBanach Theorem
75
Banach Limits
80
Runges Theorem
81
Ordered Vector Spaces
84
10 The Dual of a Quotient Space and a Subspace
86
11 Reflexive Spaces
87
12 The Open Mapping and Closed Graph Theorems
88
13 Complemented Subspaces of a Banach Space
91
14 The Principle of Uniform Boundedness
93
Locally Convex Spaces
97
2 Metrizable and Normable Locally Convex Spaces
103
3 Some Geometric Consequences of the HahnBanach Theorem
106
4 Some Examples of the Dual Space of a Locally Convex Space
112
5 Inductive Limits and the Space of Distributions
114
Weak Topologies
122
2 The Dual of a Subspace and a Quotient Space
126
3 Alaoglus Theorem
128
4 Reflexivity Revisited
129
5 Separability and Metrizability
132
The StoneCech Compactification
135
7 The KreinMilman Theorem
139
The StoneWeierstrass Theorem
143
9 The Schauder Fixed Point Theorem
147
10 The RyllNardzewski Fixed Point Theorem
149
Haar Measure on a Compact Group
152
12 The KreinSmulian Theorem
157
13 Weak Compactness
161
Linear Operators on a Banach Space
164
2 The BanachStone Theorem
169
6 The Spectrum of a Linear Operator
206
7 The Spectral Theory of a Compact Operator
212
8 Abelian Banach Algebras
216
9 The Group Algebra of a Locally Compact Abelian Group
221
CAlgebras
230
2 Abelian CAlgebras and the Functional Calculus in CAlgebras
234
3 The Positive Elements in a CAlgebra
238
4 Ideals and Quotients of CAlgebras
243
5 Representations of CAlgebras and the GelfandNaimarkSegal Construction
246
Normal Operators on Hilbert Space
253
2 The Spectral Theorem
260
3 StarCyclic Normal Operators
266
4 Some Applications of the Spectral Theorem
269
5 Topologies on B H
272
6 Commuting Operators
274
7 Abelian von Neumann Algebras
279
The Conclusion of the Saga
283
9 Invariant Subspaces for Normal Operators
288
A Complete Set of Unitary Invariants
291
Unbounded Operators
301
2 Symmetric and SelfAdjoint Operators
306
3 The Cayley Transform
314
4 Unbounded Normal Operators and the Spectral Theorem
317
5 Stones Theorem
325
6 The Fourier Transform and Differentiation
332
7 Moments
341
Fredholm Theory
345
2 Fredholm Operators
347
3 Fredholm Theory
350
4 The Essential Spectrum
356
5 The Components of J F
360
6 A Finer Analysis of the Spectrum
361
Preliminaries
367
2 Topology
369
The Dual of Lp𝜇
373
The Dual of C₀X
376
Bibliography
382
List of Symbols
389
Index
393
Copyright

Other editions - View all

Common terms and phrases

References to this book

Loop Groups
Andrew Pressley,Graeme Segal
No preview available - 1988
All Book Search results »

Bibliographic information