A first course in functional analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, NJ
Wiley
2008
|
| Schriftenreihe: | Pure and applied mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 307 S. |
| ISBN: | 9780470146194 |
Internformat
MARC
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| 245 | 1 | 0 | |a A first course in functional analysis |c S. David Promislow |
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Datensatz im Suchindex
| _version_ | 1804137665981841408 |
|---|---|
| adam_text | CONTENTS
Preface
xi
1.
Linear
Spaces
and Operators
1
1.
1 Introduction
1
1.2
Linear Spaces
2
1.3
Linear Operators
5
1.4
Passage from Finite- to Infinite-Dimensional Spaces
7
Exercises
8
2.
Normed Linear Spaces: The Basics
11
2.1
Metric Spaces
11
2.2
Norms
12
2.3
Space of Bounded Functions
18
2.4
Bounded Linear Operators
19
2.5
Completeness
2)
2.6
Comparison of Norms
30
2.7
Quotient Spaces
31
2.8
Finite-Dimensional Normed Linear Spaces
34
2.9
l! Spaces
38
2.10
Direct Products and Sums
51
2.11 Schauder
Bases
53
2.12
Fixed Points and Contraction Mappings
53
Exercises
54
3.
Major Banach Space Theorems
59
3.1
Introduction
59
3.2
Baire Category Theorem
59
3.3
Open Mappings
61
3.4
Bounded Inverses
63
iii CONTENTS
3.5
Closed Linear Operators
64
3.6
Uniform Boundedness Principle
66
Exercises
68
4.
Hubert Spaces
71
4.1
Introduction
71
4.2
Semi-Inner Products
72
4.3
Nearest Points and Convexity
77
4.4
Orthogonality
80
4.5
Linear Functionals on Hubert Spaces
86
4.6
Linear Operators on Hubert Spaces
88
4.7
Order Relation on Self-Adjoint Operators
97
Exercises
98
5.
Hahn-Banach Theorem
103
5.1
Introduction
103
5.2
Basic Version of Hahn-Banach Theorem
104
5.3
Complex Version of Hahn-Banach Theorem
105
5.4
Application to Normed Linear Spaces
107
5.5
Geometric Versions of Hahn-Banach Theorem
108
Exercises
118
6.
Duality
121
6.1
Examples of Dual Spaces
121
6.2
Adjoints
130
6.3
Double Duals and
Reflexivity
133
6.4
Weak and Weak* Convergence
136
Exercises
140
7.
Topological Linear Spaces
143
143
148
151
153
156
160
164
164
8.
The Spectrum
167
167
169
7.1
Review of General Topology
7.2
Topologies on Linear Spaces
7.3
Linear Functionals on Topological Linear Spaces
7.4
Weak Topology
7.5
Weak* Topology
7.6
Extreme Points and Krein-Milman Theorem
7.7
Operator Topologies
Exercises
The
Spectrum
8.1
Introduction
8.2
Banach Algebras
CONTENTS ix
8.3 General
Properties of the
Spectrum
170
8.4
Numerical Range
176
8.5
Spectrum of a Normal Operator
177
8.6
Functions of Operators
180
8.7
Brief Introduction to C-Algebras
183
Exercises
184
9.
Compact Operators
187
9.1
Introduction and Basic Definitions
187
9.2
Compactness Criteria in Metric Spaces
188
9.3
New Compact Operators from Old
192
9.4
Spectrum of a Compact Operator
194
9.5
Compact Self-Adjoint Operators on Hubert Spaces
197
9.6
Invariant Subspaces
201
Exercises
203
10.
Application to Integral and Differential Equations
205
10.1
Introduction
205
10.2
Integral Operators
206
10.3
Integral Equations
211
10.4
Second-Order Linear Differential Equations
214
10.5
Sturm-Liouville Problems
217
10.6
First-Order Differential Equations
223
Exercises
226
11.
Spectral Theorem for Bounded, Self-Adjoint Operators
229
11.1
Introduction and Motivation
229
11.2
Spectral Decomposition
231
11.3
Extension of Functional Calculus
235
11.4
Multiplication Operators
240
Exercises
243
Appendix A Zorn s Lemma
245
Appendix
В
Stone-Weierstrass Theorem
247
B.I Basic Theorem
247
B.2 Nonunital Algebras
250
B.3 Complex Algebras
252
Appendix
С
Extended Real Numbers and Limit Points
of Sequences
253
C.I Extended Reals
253
C.2 Limit Points of Sequences
254
x
CONTENTS
Appendix D Measure and Integration
257
D.
1
Introduction and Notation
257
D.2 Basic Properties of Measures
258
D.3 Properties of Measurable Functions
259
D.4 Integral of
a Nonnegative
Function
261
D.5 Integral of an Extended Real-Valued Function
265
D.6 Integral of a Complex-Valued Function
267
D.7 Construction of Lebesgue Measure on
Л
267
D.8 Completeness of Measures
273
D.9 Signed and Complex Measures
274
D.
10
Radon-Nikodym Derivatives
276
D.
11
Product Measures
278
D.
12
Riesz Representation Theorem
280
Appendix
E
Ţychonoff
s
Theorem
289
Symbols
293
References
297
Index
299
|
| adam_txt |
CONTENTS
Preface
xi
1.
Linear
Spaces
and Operators
1
1.
1 Introduction
1
1.2
Linear Spaces
2
1.3
Linear Operators
5
1.4
Passage from Finite- to Infinite-Dimensional Spaces
7
Exercises
8
2.
Normed Linear Spaces: The Basics
11
2.1
Metric Spaces
11
2.2
Norms
12
2.3
Space of Bounded Functions
18
2.4
Bounded Linear Operators
19
2.5
Completeness
2)
2.6
Comparison of Norms
30
2.7
Quotient Spaces
31
2.8
Finite-Dimensional Normed Linear Spaces
34
2.9
l!' Spaces
38
2.10
Direct Products and Sums
51
2.11 Schauder
Bases
53
2.12
Fixed Points and Contraction Mappings
53
Exercises
54
3.
Major Banach Space Theorems
59
3.1
Introduction
59
3.2
Baire Category Theorem
59
3.3
Open Mappings
61
3.4
Bounded Inverses
63
iii CONTENTS
3.5
Closed Linear Operators
64
3.6
Uniform Boundedness Principle
66
Exercises
68
4.
Hubert Spaces
71
4.1
Introduction
71
4.2
Semi-Inner Products
72
4.3
Nearest Points and Convexity
77
4.4
Orthogonality
80
4.5
Linear Functionals on Hubert Spaces
86
4.6
Linear Operators on Hubert Spaces
88
4.7
Order Relation on Self-Adjoint Operators
97
Exercises
98
5.
Hahn-Banach Theorem
103
5.1
Introduction
103
5.2
Basic Version of Hahn-Banach Theorem
104
5.3
Complex Version of Hahn-Banach Theorem
105
5.4
Application to Normed Linear Spaces
107
5.5
Geometric Versions of Hahn-Banach Theorem
108
Exercises
118
6.
Duality
121
6.1
Examples of Dual Spaces
121
6.2
Adjoints
130
6.3
Double Duals and
Reflexivity
133
6.4
Weak and Weak* Convergence
136
Exercises
140
7.
Topological Linear Spaces
143
143
148
151
153
156
160
164
164
8.
The Spectrum
167
167
169
7.1
Review of General Topology
7.2
Topologies on Linear Spaces
7.3
Linear Functionals on Topological Linear Spaces
7.4
Weak Topology
7.5
Weak* Topology
7.6
Extreme Points and Krein-Milman Theorem
7.7
Operator Topologies
Exercises
The
Spectrum
8.1
Introduction
8.2
Banach Algebras
CONTENTS ix
8.3 General
Properties of the
Spectrum
170
8.4
Numerical Range
176
8.5
Spectrum of a Normal Operator
177
8.6
Functions of Operators
180
8.7
Brief Introduction to C-Algebras
183
Exercises
184
9.
Compact Operators
187
9.1
Introduction and Basic Definitions
187
9.2
Compactness Criteria in Metric Spaces
188
9.3
New Compact Operators from Old
192
9.4
Spectrum of a Compact Operator
194
9.5
Compact Self-Adjoint Operators on Hubert Spaces
197
9.6
Invariant Subspaces
201
Exercises
203
10.
Application to Integral and Differential Equations
205
10.1
Introduction
205
10.2
Integral Operators
206
10.3
Integral Equations
211
10.4
Second-Order Linear Differential Equations
214
10.5
Sturm-Liouville Problems
217
10.6
First-Order Differential Equations
223
Exercises
226
11.
Spectral Theorem for Bounded, Self-Adjoint Operators
229
11.1
Introduction and Motivation
229
11.2
Spectral Decomposition
231
11.3
Extension of Functional Calculus
235
11.4
Multiplication Operators
240
Exercises
243
Appendix A Zorn's Lemma
245
Appendix
В
Stone-Weierstrass Theorem
247
B.I Basic Theorem
247
B.2 Nonunital Algebras
250
B.3 Complex Algebras
252
Appendix
С
Extended Real Numbers and Limit Points
of Sequences
253
C.I Extended Reals
253
C.2 Limit Points of Sequences
254
x
CONTENTS
Appendix D Measure and Integration
257
D.
1
Introduction and Notation
257
D.2 Basic Properties of Measures
258
D.3 Properties of Measurable Functions
259
D.4 Integral of
a Nonnegative
Function
261
D.5 Integral of an Extended Real-Valued Function
265
D.6 Integral of a Complex-Valued Function
267
D.7 Construction of Lebesgue Measure on
Л
267
D.8 Completeness of Measures
273
D.9 Signed and Complex Measures
274
D.
10
Radon-Nikodym Derivatives
276
D.
11
Product Measures
278
D.
12
Riesz Representation Theorem
280
Appendix
E
Ţychonoff
's
Theorem
289
Symbols
293
References
297
Index
299 |
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| language | English |
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| spelling | Promislow, S. David Verfasser aut A first course in functional analysis S. David Promislow Hoboken, NJ Wiley 2008 XIII, 307 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics Functional analysis Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Funktionalanalysis (DE-588)4018916-8 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016509278&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Promislow, S. David A first course in functional analysis Functional analysis Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4018916-8 (DE-588)4123623-3 |
| title | A first course in functional analysis |
| title_auth | A first course in functional analysis |
| title_exact_search | A first course in functional analysis |
| title_exact_search_txtP | A first course in functional analysis |
| title_full | A first course in functional analysis S. David Promislow |
| title_fullStr | A first course in functional analysis S. David Promislow |
| title_full_unstemmed | A first course in functional analysis S. David Promislow |
| title_short | A first course in functional analysis |
| title_sort | a first course in functional analysis |
| topic | Functional analysis Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Functional analysis Funktionalanalysis Lehrbuch |
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