A Course in Functional Analysis

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Springer Science & Business Media, 17.04.2013 - 406 Seiten
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.
 

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Inhalt

CHAPTER
1
2 Orthogonality
7
3 The Riesz Representation Theorem
11
4 Orthonormal Sets of Vectors and Bases
14
5 Isomorphic Hilbert Spaces and the Fourier Transform for the Circle
19
6 The Direct Sum of Hilbert Spaces
24
CHAPTER II
26
2 The Adjoint of an Operator
31
10 The RyllNardzewski FixedPoint Theorem
155
Haar Measure on a Compact Group
158
12 The KreinSmulian Theorem
163
13 Weak Compactness
167
CHAPTER VI
170
2 The BanachStone Theorem
175
3 Compact Operators
177
4 Invariant Subspaces
182

3 Projections and Idempotents Invariant and Reducing Subspaces
37
4 Compact Operators
41
5 The Diagonalization of Compact SelfAdjoint Operators
47
SturmLiouville Systems
50
7 The Spectral Theorem and Functional Calculus for Compact Normal Operators
55
8 Unitary Equivalence for Compact Normal Operators
61
CHAPTER III
65
2 Linear Operators on Normed Spaces
70
3 FiniteDimensional Normed Spaces
71
4 Quotients and Products of Normed Spaces
73
5 Linear Functionals
76
6 The HahnBanach Theorem
80
Banach Limits
84
Runges Theorem
86
Ordered Vector Spaces
88
10 The Dual of a Quotient Space and a Subspace
91
11 Reflexive Spaces
92
12 The Open Mapping and Closed Graph Theorems
93
13 Complemented Subspaces of a Banach Space
97
14 The Principle of Uniform Boundedness
98
CHAPTER IV
102
2 Metrizable and Normable Locally Convex Spaces
108
3 Some Geometric Consequences of the HahnBanach Theorem
111
4 Some Examples of the Dual Space of a Locally Convex Space
117
5 Inductive Limits and the Space of Distributions
119
CHAPTER V
127
2 The Dual of a Subspace and a Quotient Space
131
3 Alaoglus Theorem
134
4 Reflexivity Revisited
135
5 Separability and Metrizability
138
The StoneČech Compactification
140
7 The KreinMilman Theorem
145
The StoneWeierstrass Theorem
149
9 The Schauder FixedPoint Theorem
153
5 Weakly Compact Operators
187
Banach Algebras and Spectral Theory for Operators on a Banach Space 1 Elementary Properties and Examples
191
2 Ideals and Quotients
195
3 The Spectrum
199
4 The Riesz Functional Calculus
203
5 Dependence of the Spectrum on the Algebra
210
6 The Spectrum of a Linear Operator
213
7 The Spectral Theory of a Compact Operator
219
8 Abelian Banach Algebras
222
9 The Group Algebra of a Locally Compact Abelian Group
228
CHAPTER VIII
237
2 Abelian CAlgebras and the Functional Calculus in CAlgebras
242
3 The Positive Elements in a CAlgebra
245
4 Ideals and Quotients for CAlgebras
250
5 Representations of CAlgebras and the GelfandNaimarkSegal Construction
254
CHAPTER IX
261
2 The Spectral Theorem
268
3 StarCyclic Normal Operators
275
4 Some Applications of the Spectral Theorem
278
5 Topologies on 33
281
6 Commuting Operators
283
7 Abelian von Neumann Algebras
288
The Conclusion of the Saga
292
9 Invariant Subspaces for Normal Operators
297
A Complete Set of Unitary Invariants
299
CHAPTER X
310
2 Symmetric and SelfAdjoint Operators
316
3 The Cayley Transform
323
4 Unbounded Normal Operators and the Spectral Theorem
326
CHAPTER XI
353
APPENDIX
375
Bibliography
390
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