A first course in functional analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton
CRC Press
[2017]
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xv, 240 Seiten |
| ISBN: | 9781498771610 |
Internformat
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| 100 | 1 | |a Shalit, Orr Moshe |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a A first course in functional analysis |c Orr Moshe Shalit, Technion - Israel Institute of Technology Haifa, Israel |
| 264 | 1 | |a Boca Raton |b CRC Press |c [2017] | |
| 264 | 4 | |c © 2017 | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Preface xi
1 Introduction and the Stone-Weierstrass theorem 1
1.1 Background and motivation..................................... 1
1.2 The Weierstrass approximation theorem......................... 3
1.3 The Stone-Weierstrass theorem................................. 5
1.4 The Stone-Weierstrass theorem over the complex numbers . 8
1.5 Concluding remarks............................................ 9
1.6 Additional exercises......................................... 10
2 Hilbert spaces 13
2.1 Background and motivation.................................... 13
2.2 The basic definitions........................................ 14
2.3 Completion................................................... 19
2.4 The space of Lebesgue square integrable functions......... 21
2.5 Additional exercises ........................................ 25
3 Orthogonality, projections, and bases 29
3.1 Orthogonality ............................................... 29
3.2 Orthogonal projection and orthogonal decomposition .... 30
3.3 Orthonormal bases............................................ 33
3.4 Dimension and isomorphism ................................... 38
3.5 The Gram-Schmidt process..................................... 40
3.6 Additional exercises......................................... 40
4 Fourier series 45
4֊
4.1 Fourier series in L2 ........................................ 45
4.2 Pointwise convergence of Fourier series (Dirichlet’s theorem) 50
4.3 Fejér’s theorem.............................................. 52
4.4 *Proof of Dirichlet’s theorem................................ 54
4.5 Additional exercises...................................... 55
Vll
Vlll
Contents
5 Bounded linear operators on Hilbert space 61
5.1 Bounded operators............................................ 61
5.2 Linear functionals and the Riesz representation theorem ... 64
5.3 *The Dirichlet problem for the Laplace operator.............. 65
5.4 The adjoint of a bounded operator............................ 70
5.5 Special classes of operators................................. 72
5.6 Matrix representation of operators........................... 73
5.7 Additional exercises......................................... 75
6 Hilbert function spaces 83
6.1 Reproducing kernel Hilbert spaces............................ 83
6.2 *The Bergman space .......................................... 87
6.3 * Additional topics in Hilbert function space theory ........ 91
6.4 Additional exercises........................................ 100
7 Banach spaces 103
7.1 Banach spaces .............................................. 103
7.2 Bounded operators........................................... 108
7.3 The dual space.............................................. 110
7.4 *Topological vector spaces ................................. 113
7.5 Additional exercises........................................ 114
8 The algebra of bounded operators on a Banach space 119
8.1 The algebra of bounded operators ........................... 119
8.2 An application to ergodic theory............................ 120
8.3 Invertible operators and inverses........................... 124
8.4 isomorphisms................................................ 129
8.5 Additional exercises........................................ 131
9 Compact operators 137
9.1 Compact operators ........................................ 137
9.2 The spectrum of a compact operator ......................... 139
9.3 Additional exercises........................................ 142
10 Compact operators on Hilbert space 145
10.1 Finite rank operators on Hilbert space...................... 145
10.2 The spectral theorem for compact self-adjoint operators ... 146
10.3 The spectral theorem for compact normal operators........... 150
10.4 The functional calculus for compact normal operators .... 152
A
10.5 Additional exercises........................................ 155
Contents ix
11 Applications of the theory of compact operators 161
11.1 Integral equations ...................................... 161
11.2 *Functional equations.................................... 171
11.3 Additional exercises..................................... 174
12 The Fourier transform 177
12.1 The spaces TP(R), p G [l,oo)............................. 177
12.2 The Fourier transform on Z/1(R) 182
12.3 The Fourier transform on L2(R) 189
12.4 *Shannon’s sampling theorem ............................. 193
12.5 *The multivariate Fourier transforms .................... 194
12.6 Additional exercises..................................... 197
13 *The Hahn-Banach theorems 201
13.1 The Hahn-Banach theorems ................................ 201
13.2 The dual space, the double dual, and duality............. 209
13.3 Quotient spaces ......................................... 211
13.4 Additional excercises ................................... 214
Appendix A Metric and topological spaces 217
A.l Metric spaces............................................. 217
A. 2 Completeness............................................ 221
A.3 Topological spaces........................................ 225
A.4 The Arzelà-Ascoli theorem ................................ 230
Symbol Description 233
Bibliography 235
Index
237
|
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| spelling | Shalit, Orr Moshe Verfasser aut A first course in functional analysis Orr Moshe Shalit, Technion - Israel Institute of Technology Haifa, Israel Boca Raton CRC Press [2017] © 2017 xv, 240 Seiten txt rdacontent n rdamedia nc rdacarrier Functional analysis Textbooks 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 Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029717130&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shalit, Orr Moshe A first course in functional analysis Functional analysis Textbooks 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_full | A first course in functional analysis Orr Moshe Shalit, Technion - Israel Institute of Technology Haifa, Israel |
| title_fullStr | A first course in functional analysis Orr Moshe Shalit, Technion - Israel Institute of Technology Haifa, Israel |
| title_full_unstemmed | A first course in functional analysis Orr Moshe Shalit, Technion - Israel Institute of Technology Haifa, Israel |
| title_short | A first course in functional analysis |
| title_sort | a first course in functional analysis |
| topic | Functional analysis Textbooks Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Functional analysis Textbooks Funktionalanalysis Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029717130&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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