Functional analysis: an elementary introduction
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
2014
|
| Schriftenreihe: | Graduate studies in mathematics
156 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XVIII, 372 S. graph. Darst. |
| ISBN: | 9780821891711 |
Internformat
MARC
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| 003 | DE-604 | ||
| 005 | 20141121 | ||
| 007 | t | ||
| 008 | 141014s2014 d||| |||| 00||| eng d | ||
| 016 | 7 | |a 785606157 |2 DE-101 | |
| 020 | |a 9780821891711 |c alk. paper |9 978-0-8218-9171-1 | ||
| 035 | |a (OCoLC)891769833 | ||
| 035 | |a (DE-599)GBV785606157 | ||
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| 084 | |a MAT 460f |2 stub | ||
| 100 | 1 | |a Haase, Markus |d 1970- |e Verfasser |0 (DE-588)1060737418 |4 aut | |
| 245 | 1 | 0 | |a Functional analysis |b an elementary introduction |c Markus Haase |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c 2014 | |
| 300 | |a XVIII, 372 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 156 | |
| 650 | 0 | 7 | |a Funktionalanalysis |0 (DE-588)4018916-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Funktionalanalysis |0 (DE-588)4018916-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a Graduate studies in mathematics |v 156 |w (DE-604)BV009739289 |9 156 | |
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| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027556928&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-027556928 | ||
Datensatz im Suchindex
| DE-BY-862_location | 2000 |
|---|---|
| DE-BY-FWS_call_number | 2000/SK 600 H112 |
| DE-BY-FWS_katkey | 559520 |
| DE-BY-FWS_media_number | 083000514896 |
| _version_ | 1806179257808846848 |
| adam_text | Contents
Preface
xiii
Chapter
1.
Inner Product Spaces
1
§1.1.
Inner Products
3
§1.2.
Orthogonality
6
§1.3.
The Trigonometric System
10
Exercises
11
Chapter
2.
Normed Spaces
15
§2.1.
The Cauchy-Schwarz Inequality and the Space £2
15
§2.2.
Norms
18
§2.3.
Bounded Linear Mappings
21
§2.4.
Basic Examples
23
§2.5.
*The ^-Spaces
(1 <
ρ
<
oo)
28
Exercises
31
Chapter
3.
Distance and Approximation
37
§3.1.
Metric Spaces
37
§3.2.
Convergence
39
§3.3.
Uniform, Pointwise and (Square) Mean Convergence
41
§3.4.
The Closure of a Subset
47
Exercises
50
Chapter
4.
Continuity and Compactness
55
§4.1.
Open and Closed Sets
55
vii
§4.2.
Continuity
58
§4.3.
Sequential Compactness
64
§4.4.
Equivalence of Norms
66
§4.5.
*Separability and General Compactness
71
Exercises
74
Chapter
5.
Banach Spaces
79
§5.1.
Cauchy Sequences and Completeness
79
§5.2.
Hubert Spaces
81
§5.3.
Banach Spaces
84
§5.4.
Series in Banach Spaces
86
Exercises
88
Chapter
6.
*The Contraction Principle
93
§6.1.
Banach s Contraction Principle
94
§6.2.
Application: Ordinary Differential Equations
95
§6.3.
Application: Google s PageRank
98
§6.4.
Application: The Inverse Mapping Theorem
100
Exercises
104
Chapter
7.
The Lebesgue Spaces
107
§7.1.
The Lebesgue Measure
110
§7.2.
The Lebesgue Integral and the Space l}(X)
113
§7.3.
Null Sets
115
§7.4.
The Dominated Convergence Theorem
118
§7.5.
The Spaces IP(X) with
1 <
ρ
<
oo
121
Advice for the Reader
125
Exercises
126
Chapter
8.
Hubert Space Fundamentals
129
§8.1.
Best Approximations
129
§8.2.
Orthogonal Projections
133
§8.3.
The
Riesz-Fréchet
Theorem
135
§8.4.
Orthogonal Series and Abstract Fourier Expansions
137
Exercises
141
Chapter
9.
Approximation Theory and Fourier Analysis
147
§9.1.
Lebesgue s Proof of
Weierstrass
Theorem
149
§9.2.
Truncation
151
§9.3.
Classical Fourier Series
156
§9.4.
Fourier Coefficients of ^-Functions
161
§9.5.
The Riemann-Lebesgue Lemma
162
§9.6.
*The Strong Convergence Lemma and
Fejér s
Theorem
164
§9.7.
*Extension of a Bounded Linear Mapping
168
Exercises
172
Chapter
10.
Sobolev Spaces and the
Poisson
Problem
177
§10.1.
Weak Derivatives
177
§10.2.
The Fundamental Theorem of Calculus
179
§10.3.
Sobolev Spaces
182
§10.4.
The Variational Method for the
Poisson
Problem
184
§10.5.
*Poisson s Problem in Higher Dimensions
187
Exercises
188
Chapter
11.
Operator Theory I
193
§11.1.
Integral Operators and Fubini s Theorem
193
§11.2.
The Dirichlet Laplacian and Hilbert-Schmidt Operators
196
§11.3.
Approximation of Operators
199
§11.4.
The Neumann Series
202
Exercises
205
Chapter
12.
Operator Theory II
211
§12.1.
Compact Operators
211
§12.2.
Adjoints
of Hubert Space Operators
216
§12.3.
*The Lax-Milgram Theorem
219
§12.4.
*Abstract Hubert-Schmidt Operators
221
Exercises
226
Chapter
13.
Spectral Theory of Compact Self-Adjoint Operators
231
§13.1.
Approximate Eigenvalues
231
§13.2.
Self-Adjoint Operators
234
§13.3.
The Spectral Theorem
236
§13.4.
*The General Spectral Theorem
240
Exercises
241
Chapter
14.
Applications of the Spectral Theorem
247
§14.1.
The Dirichlet Laplacian
247
§14.2.
The
Schrödinger
Operator
249
§14.3.
An Evolution Equation
252
§14.4.
*The Norm of the Integration Operator
254
§14.5.
*The Best Constant in the
Poincaré
Inequality
256
Exercises
257
Chapter
15.
Baire s Theorem and Its Consequences
261
§15.1.
Baire s Theorem
261
§15.2.
The Uniform Boundedness Principle
263
§15.3.
Nonconvergence of Fourier Series
266
§15.4.
The Open Mapping Theorem
267
§15.5.
Applications with a Look Back
271
Exercises
274
Chapter
16.
Duality and the Hahn-Banach Theorem
277
§16.1.
Extending Linear Functionals
278
§16.2.
Elementary Duality Theory
284
§16.3.
Identification of Dual Spaces
289
§16.4.
*The Riesz Representation Theorem
295
Exercises
299
Historical Remarks
305
Appendix A. Background
311
§A.l. Sequences and Subsequences
311
§A.2. Equivalence Relations
312
§A.3. Ordered Sets
314
§A.4. Countable and Uncountable Sets
316
§A.5. Real Numbers
316
§A.6. Complex Numbers
321
§A.7. Linear Algebra
322
§A.8. Set-theoretic Notions
329
Appendix B. The Completion of a Metric Space
333
§B.l. Uniqueness of a Completion
334
§B.2. Existence of a Completion
335
§B.3. The Completion of a Normed Space
337
Exercises
338
Appendix
С.
Bernstein s Proof of
Weierstrass
Theorem
339
Appendix D. Smooth Cutoff Functions
343
Appendix E. Some Topics from Fourier Analysis
345
§E.l. Plancherel s Identity
346
§E.2. The Fourier Inversion Formula
347
§E.3. The Carlson-Beurling Inequality
348
Exercises
349
Appendix F. General
Orthonormal
Systems
351
§F.l. Unconditional Convergence
351
§F.2. Uncountable
Orthonormal
Bases
353
Bibliography
355
Symbol Index
359
Subject Index
361
Author Index
371
This book introduces functional analysis at an elementary level without assuming any
background in real analysis, for example on metric spaces or Lebesgue integration.
It focuses on concepts and methods relevant in applied contexts such as variational
methods on Hubert spaces, Neumann series, eigenvalue expansions for compact self-
adjoint operators, weak differentiation and Sobolev spaces on intervals, and model
applications to differential and integral equations. Beyond that, the final chapters on
the uniform boundedness theorem, the open mapping theorem and the Hahn-Banach
theorem provide a stepping-stone to more advanced texts.
The exposition is clear and rigorous, featuring full and detailed proofs. Many examples
illustrate the new notions and results. Each chapter concludes with a large collection
of exercises, some of which are referred to in the margin of the text, tailor-made in
order to guide the student digesting the new material. Optional sections and chapters
supplement the mandatory parts and allow for modular teaching spanning from basic
to honors track level.
|
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| classification_tum | MAT 460f |
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| discipline | Mathematik |
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| id | DE-604.BV042116613 |
| illustrated | Illustrated |
| indexdate | 2024-08-01T12:06:08Z |
| institution | BVB |
| isbn | 9780821891711 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027556928 |
| oclc_num | 891769833 |
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| physical | XVIII, 372 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spellingShingle | Haase, Markus 1970- Functional analysis an elementary introduction Graduate studies in mathematics Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4018916-8 |
| title | Functional analysis an elementary introduction |
| title_auth | Functional analysis an elementary introduction |
| title_exact_search | Functional analysis an elementary introduction |
| title_full | Functional analysis an elementary introduction Markus Haase |
| title_fullStr | Functional analysis an elementary introduction Markus Haase |
| title_full_unstemmed | Functional analysis an elementary introduction Markus Haase |
| title_short | Functional analysis |
| title_sort | functional analysis an elementary introduction |
| title_sub | an elementary introduction |
| topic | Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027556928&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027556928&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
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Inhaltsverzeichnis
THWS Schweinfurt Zentralbibliothek Lesesaal
| Signatur: |
2000 SK 600 H112 |
|---|---|
| Exemplar 1 | ausleihbar Verfügbar Bestellen |