Lectures and exercises on functional analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Providence, R.I.
American Math. Soc.
2006
|
| Series: | Translations of mathematical monographs
233 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | XVII, 468 S. |
| ISBN: | 0821840983 9780821840986 |
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| 240 | 1 | 0 | |a Lektsii po funktsionalnomu analizu |
| 245 | 1 | 0 | |a Lectures and exercises on functional analysis |c A. Ya. Helemskii |
| 264 | 1 | |a Providence, R.I. |b American Math. Soc. |c 2006 | |
| 300 | |a XVII, 468 S. | ||
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| 490 | 1 | |a Translations of mathematical monographs |v 233 | |
| 500 | |a Aus dem Russ. übers. | ||
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Record in the Search Index
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| adam_text | Contents
Preface xi
Chapter 0. Foundations: Categories and the Like 1
§1. On sets and linear and metric spaces 2
§2. Topological spaces 9
§3. Categories; first examples 20
§4. Isomorphisms. The problem of classification of objects and
morphisms 25
§5. Other classes of morphisms 32
§6. A sample of a category theoretic construction: The (co)product 37
§7. Functors 45
Chapter 1. Normed Spaces and Bounded Operators
( Waiting for Completeness ) 55
§1. Prenormed and normed spaces. Examples 55
§2. Inner products and near Hilbert spaces 66
§3. Bounded operators: First acquaintance, examples 76
§4. Topological and categorical properties of bounded operators 82
§5. Some types of operators and operator constructions. Projections 93
§6. Functionals and the Hahn Banach theorem 100
§7. Invitation to quantum functional analysis 113
Chapter 2. Banach Spaces and Their Advantages 125
§1. What lies on the surface 125
vii
§2. Categories of Banach and Hilbert spaces. Classification and the
Riesz Fischer theorem 133
§3. Theorem on the orthogonal complement and around it 140
§4. Open mapping principle and uniform boundedness principle 149
§5. Banach adjointness functor and other categorical questions 155
§6. Completion 167
§7. Algebraic and Banach tensor products 173
§8. Hilbert tensor product 186
Chapter 3. From Compact Spaces to Fredholm Operators 191
§1. Compact spaces and relevant functional spaces 191
§2. Compact metric spaces and total boundedness 201
§3. Compact operators: General properties and examples 209
§4. Compact operators between Hilbert spaces 215
§5. Fredholm operators and the index 232
Chapter 4. Polynormed Spaces, Weak Topologies, and Generalized
Functions 245
§1. Polynormed spaces 245
§2. Weak topologies 260
§3. Spaces of test functions and generalized functions 276
§4. Generalized derivatives and the structure of generalized
functions 288
Chapter 5. At the Gates of Spectral Theory 297
§1. Spectra of operators and their classification. Examples 297
§2. Something from algebra: Algebras 304
§3. Banach algebras and spectra of their elements.
Fredholm operators revisited 311
Chapter 6. Hilbert Adjoint Operators and the Spectral Theorem 327
§1. Hilbert adjointness: First information 327
§2. Selfadjoint operators and their spectra. Hilbert Schmidt
theorem 336
§3. An overview: Involutive algebras, C* algebras, and
von Neumann algebras 347
§4. Continuous functional calculus and positive operators 360
§5. The spectral theorem as an operator valued
Riemann Stiltjes integral 371
§6. Borel calculus and the spectral theorem as an operator valued
Lebesgue integral 382
§7. Geometric form of the spectral theorem: Models and
classification 396
§8. Proof of the final form of the spectral theorem 405
Chapter 7. Fourier Transform 413
§1. Classical Fourier transform 413
§2. Convolution. Fourier transform as a homomorphism 422
§3. Fourier transform of test functions and of generalized functions 433
§4. Fourier transform of square integrable functions 441
§5. A little about harmonic analysis on groups 448
Bibliography 455
Index 461
|
| adam_txt |
Contents
Preface xi
Chapter 0. Foundations: Categories and the Like 1
§1. On sets and linear and metric spaces 2
§2. Topological spaces 9
§3. Categories; first examples 20
§4. Isomorphisms. The problem of classification of objects and
morphisms 25
§5. Other classes of morphisms 32
§6. A sample of a category theoretic construction: The (co)product 37
§7. Functors 45
Chapter 1. Normed Spaces and Bounded Operators
("Waiting for Completeness") 55
§1. Prenormed and normed spaces. Examples 55
§2. Inner products and near Hilbert spaces 66
§3. Bounded operators: First acquaintance, examples 76
§4. Topological and categorical properties of bounded operators 82
§5. Some types of operators and operator constructions. Projections 93
§6. Functionals and the Hahn Banach theorem 100
§7. Invitation to quantum functional analysis 113
Chapter 2. Banach Spaces and Their Advantages 125
§1. What lies on the surface 125
vii
§2. Categories of Banach and Hilbert spaces. Classification and the
Riesz Fischer theorem 133
§3. Theorem on the orthogonal complement and around it 140
§4. Open mapping principle and uniform boundedness principle 149
§5. Banach adjointness functor and other categorical questions 155
§6. Completion 167
§7. Algebraic and Banach tensor products 173
§8. Hilbert tensor product 186
Chapter 3. From Compact Spaces to Fredholm Operators 191
§1. Compact spaces and relevant functional spaces 191
§2. Compact metric spaces and total boundedness 201
§3. Compact operators: General properties and examples 209
§4. Compact operators between Hilbert spaces 215
§5. Fredholm operators and the index 232
Chapter 4. Polynormed Spaces, Weak Topologies, and Generalized
Functions 245
§1. Polynormed spaces 245
§2. Weak topologies 260
§3. Spaces of test functions and generalized functions 276
§4. Generalized derivatives and the structure of generalized
functions 288
Chapter 5. At the Gates of Spectral Theory 297
§1. Spectra of operators and their classification. Examples 297
§2. Something from algebra: Algebras 304
§3. Banach algebras and spectra of their elements.
Fredholm operators revisited 311
Chapter 6. Hilbert Adjoint Operators and the Spectral Theorem 327
§1. Hilbert adjointness: First information 327
§2. Selfadjoint operators and their spectra. Hilbert Schmidt
theorem 336
§3. An overview: Involutive algebras, C* algebras, and
von Neumann algebras 347
§4. Continuous functional calculus and positive operators 360
§5. The spectral theorem as an operator valued
Riemann Stiltjes integral 371
§6. Borel calculus and the spectral theorem as an operator valued
Lebesgue integral 382
§7. Geometric form of the spectral theorem: Models and
classification 396
§8. Proof of the final form of the spectral theorem 405
Chapter 7. Fourier Transform 413
§1. Classical Fourier transform 413
§2. Convolution. Fourier transform as a homomorphism 422
§3. Fourier transform of test functions and of generalized functions 433
§4. Fourier transform of square integrable functions 441
§5. A little about harmonic analysis on groups 448
Bibliography 455
Index 461 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T14:49:19Z |
| indexdate | 2024-07-09T20:39:45Z |
| institution | BVB |
| isbn | 0821840983 9780821840986 |
| language | English Russian |
| lccn | 2005053605 |
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| publisher | American Math. Soc. |
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| spelling | Chelemskij, Aleksandr Ja. 1943- Verfasser (DE-588)143562983 aut Lektsii po funktsionalnomu analizu Lectures and exercises on functional analysis A. Ya. Helemskii Providence, R.I. American Math. Soc. 2006 XVII, 468 S. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 233 Aus dem Russ. übers. Functional analysis Operator theory Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 s DE-604 Translations of mathematical monographs 233 (DE-604)BV000002394 233 DE-605 pdf/application http://www.gbv.de/dms/hbz/toc/ht014837818.pdf 2008-11-15 Inhaltsverzeichnis http://www.loc.gov/catdir/toc/fy0703/2005053605.html kostenfrei Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014822545&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chelemskij, Aleksandr Ja. 1943- Lectures and exercises on functional analysis Translations of mathematical monographs Functional analysis Operator theory Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4018916-8 |
| title | Lectures and exercises on functional analysis |
| title_alt | Lektsii po funktsionalnomu analizu |
| title_auth | Lectures and exercises on functional analysis |
| title_exact_search | Lectures and exercises on functional analysis |
| title_exact_search_txtP | Lectures and exercises on functional analysis |
| title_full | Lectures and exercises on functional analysis A. Ya. Helemskii |
| title_fullStr | Lectures and exercises on functional analysis A. Ya. Helemskii |
| title_full_unstemmed | Lectures and exercises on functional analysis A. Ya. Helemskii |
| title_short | Lectures and exercises on functional analysis |
| title_sort | lectures and exercises on functional analysis |
| topic | Functional analysis Operator theory Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Functional analysis Operator theory Funktionalanalysis |
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