Methods of Hilbert spaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Polish |
| Published: |
Warszawa
PWN-Polish Scientific Publ.
1972
|
| Edition: | 2. ed., rev. |
| Series: | Monografie matematyczne
45 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Poln. übers. |
| Physical Description: | 553 S. |
Staff View
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| 100 | 1 | |a Maurin, Krzysztof |d 1923- |e Verfasser |0 (DE-588)1014360323 |4 aut | |
| 240 | 1 | 0 | |a Metody przestrzeni Hilberta |
| 245 | 1 | 0 | |a Methods of Hilbert spaces |c Krzysztof Maurin |
| 250 | |a 2. ed., rev. | ||
| 264 | 1 | |a Warszawa |b PWN-Polish Scientific Publ. |c 1972 | |
| 300 | |a 553 S. | ||
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| 490 | 1 | |a Monografie matematyczne |v 45 | |
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
Preface to the Polish edition 7
Prefect to the English edition 11
CHAPTER I
Metric spaces, Banach spaces, nitary spaces and Hubert space*
1. Preliminary concepts 21
2. Hermitian forms in unitary spaces 21
3. Examples of unitary spaces. Real spaces 23
4. Definition of a Hilbert space. Completion of unitary spaces 24
5. The space of continuous maps C(E,F) 28
6. Completion of unitary spaces . . . 39
Exercises and problems 33
CHAPTER II
Geometry of Hilbert space
1. Linear dependence of vectors. The n dimensional space 36
2. Infinitely dimensional Hilbert spaces 37
3. Orthogonality. Complete sets . 39
4. Theorem of Beppo Levi. Orthogonal decomposition of a Hilbert space. . . 40
5. Bases in a Hilbert space 43
6. Deficiency of a subspace or a hyperplane. The general form of linear functionate 47
7. The Hahn Banach theorem SO
Exercises and problems 53
chapter in
Locally conrex vector spaces. The dosed paph theorem. The kernel theorem
Introduction • 55
1. The Baire theorem . . . . • 56
2. Linear topological spaces. The closed graph theorem for linear metric spaces 57
3. Some other theorems of Banach 60
4. Locally convex linear topological spaces 61
5. Bilinear forms on ^ spaces . ., 68
6. Linear maps on I.e. spaces. Further properties of bomological spaces. Bar¬
relled spaces 70
7. Formation of new I.e. spaces ?3
8. Examples of locally convex spaces 76
9. The closed graph theorem •¦ 78
14 Contents
10. Tensor products. The kernel theorem 82
11. Closable maps. The index of a map 87
Exercises. Supplementary remarks 89
Appendix 89
CHAPTER IV
Hermitian operators. Spectral decomposition of Hermitian operators
1. Adjoint operators 95
2. Examples of Hermitian operators 97
3. Spectral theorem (finite dimensional case). 98
4. Spectral theorem in an arbitrary Hilbert space 100
Exercises 104
CHAPTER V
Symmetric and self adjoint operators. Extensions of symmetric operators to self adjoint operators
1. Unbounded operators 105
2. The operator A*. Examples of adjoint operators 106
3. Theorems of v. Neumann ¦ 108
4. Symmetric and semibounded operators. Friedrichs extension 110
5. Spectral representation of semibounded operators 112
6. Functions of a semibounded operator. The Stone v. Neumann operator calculus 113
7. The spectrum of a self adjoint operator 118
8. General remarks on differential operators. Examples of symmetric differential
operators 120
Appendix 123
CHAPTER VI
Isometric and unitary operators. Self adjoint extension of symmetric operators.
Spectral theorem for self adjoint operators
1. Isometric operators 124
2. Unitary extensions of isometric operators 126
3. Cayley transformation. Theorem of v. Neumann 127
4. Examples of differential operators that have self adjoint extensions. A lemma ,
of L. Maurin 129
5. Spectral theorem for self adjoint operators 131
Exercise 133
CHAPTER VII j
Compact operators. Riesz theory of the linear equation with compact operator. Examples
1. Continuous finite dimensional operators. Definition of compact operators. . 134
2. Properties of compact operators 137
3. Riesz theory of linear equations of the second kind 140
4. The spectrum of a compact operator. The Hilbert Schmidt theorem. . . . 144
5. The spectral theorem of Rellich 145
6. Weak convergence in a Hilbert space. A theorem of Riesz 148
7. Hilbert Schmidt operators 154
8. Hilbert Schmidt maps 157
9. Nuclear maps. Nuclear spaces 159
Contents 15
CHAPTER VIII
Commutative Banach algebras. The Gelfand Naimark theorem. Maximal C* algebras. Applications
1. Algebra of bounded operators in a Banach space. Spectrum of an operator 165
2. Gelfand s theory of maximal ideals 169
3. Commutative C* algebras. The Gelfand Naimark theorem 176
4. Maximal commutative C* algebras of operators in a Hilbert space .... 181
5. Completeness of a commuting system of operators 184
Exercises 185
CHAPTER IX
Direct integrals of Hilbert spaces. Two proofs of spectral theorem. V. Neumann diagonal
representation theorem for C* algebras. Measurable fields of Hilbert spaces
1. The Gelfand Naimark theorem 187
2. The proof of spectral theorem based on Gelfand Naimark theorem .... 189
3. Direct integral of Hilbert spaces 191
4. The complete spectral theorem of v. Neumann 193
5. Banach algebras without identity. Another proof of the complete spectral theorem 199
6. Measurable fields of Hilbert spaces 206
Exercises 208
chapter x
One parameter semigroups of linear operations (the theory of YosJda). Stone s theorem.
Garding theorem on the representation of lie groups
1. Heuristic considerations. Basic concepts 209
2. Semigroups of operators and their/infinitesimal generators 211
3. Theorem of Hille Yosida . . . . 214
4. One parameter groups of unitary operators. The theorem of Stone .... 219
5. Supplementary remarks 221
6. Garding theorem on representations of Lie groups 222
7. Unitary representation of Lie groups. Representations of Lie algebras. Relations
with the quantum theory 225
Exercises 234
Supplementary information 235
CHAPTER XI
Fundamental theorem on weak solutions of elliptic systems. Green fraction
, 1. The fundamental theorem 237
2. Consequences of the fundamental theorem on weak solutions 242
3. Green function and its properties 247
* Exercises 246
CHAPTER XII
The method of self adjoint extensions
1. Dirichlet problem 247
2. Eigenvalue problems 249
3. On the theory of ViSik 251
¦i
1
16 Contents
CHAPTER XIII
The boundary value and eigenvalue problems for the general elliptic operators of arbitrary order.
The strongly elliptic systems. The Galerkin method
1. The general Dirichlet problem 256
2. The elliptic operators are semi bounded 256
3. The generalized Dirichlet integral . . 259
4. The solution of the Dirichlet problem 263
5. The pairs of equations with adjoint operators 266
6. A generalization of the Neumann problem 267
7. The Galerkin method 268
8. The eigenvalue problem. The Green transformation . 270
Problems and exercises 273
CHAPTER XIV
Ehrling inequalities. Compactness of embedding operator. Sobolev lemma. Weak semi continuity
of functknab occurring in calculus of variations for multiple integrals
1. In which we present the problem 276
2. The Ehrling regions . . 276
3. The Ehrling inequalities 279
4. Compact subsets in L*. The compactness of the embedding ljit 284
5. The spaces 2T_i( k 0 288
6. The fundamental inequalities 290
7. Vibration problems. Fundamental inequalities 291
8. Semi continuous functions _ 295
9. Semi continuous quadratic forms and Legendre forms 295
10. The quadratic forms occurring in the calculus of variations 299
11. Functionate occurring in the theory of elastic plates. Differentiability of
elements giving absolute minima . . 302
12. Coercivity of integro differential forms 304
13. Small perturbations of boundary value problems 308
14. The resolvent for elliptic boundary problems 312
Exercises . . 313
CHAPTER XV
The orthogonal projection method (Dirichlet principle) and its connection with methods
of Treffiz, Rite, Schwarz and Pomcare
1. Heuristic lead 315
2. The space § and its orthogonal decomposition Z@U 316
3. Trefftz method and Ritz method. . . 318
4. Classical construction of Trefftz and Ritz sequences 318
5. Self adjoint systems of elliptic equations 320
6. The method of orthogonal projection for general non negative elliptic forms.
Other approximate methods; methods of Schwarz and Poincare 321
Exercises 326
CHAPTER XVI
The Bochner theorem on analytic embedding of compact Riemannian manifolds
into the Euclidean spaces
1. The idea of the proof 327
2. The spectrum of the operator Ae 328
3. The uniform approximation of a differentiate function by analytic functions 329
Contents 17
CHAPTER XVII
General theory of eigenfunction expansions. The spectral theory of generalized kernels
Introduction 332
1. The fundamental theorem and its consequences 333
2. For m m/2 the embeddings Hm(Qn) *H (Qn) are H. S 336
3. Theorems on eigenfunction expansions for operators of classical analysis 337
4. Eigenfunction expansions of dynamic systems 341
5. A construction of the nuclear space 0 for a countable family (Aj) .... 342
6. Eigenfunction expansions for differential operators on a Lie group and their
representants 345
7. The spectral representation of the generalized kernels 347
8. The theorems on expansions in eigenfunctiona related with Carleman kernels 354
9. Linear elliptic operators of an arbitrary order 361
10. On eigenfanction expansions for arbitrary self adjoint operators in £*(fin) 364
11. The ordinary differential operators. Spectral matrices. The Fourier transform
in I?, the Plancherel theory 367
12. The asymptotic properties of the spectral function for a semi bounded self
adjoint extension of elliptic operators 374
CHAPTER XVIII
The Fourier method
1. Heuristic considerations 378
2. The operator version and its solution 380
3. Differentiability of the generalized solution 383
4. Solution of the mixed problem for parabolic systems of type du/dt = — Axu 386
1 du
5. The mixed problem and the Cauchy problem for equations of type —== —
y — 1 dt
= Axu (equations of quantum mechanics) 386
6. Mixed conditions for inhomogeneous equations 388
7. Concluding remarks 389
Exercises 390
CHAPTER XIX
Theory of harmonic fields
1. Definition of harmonic fields 391
2. Kodaira theorems 395
3. The equation da/dt = Aa and the method of orthogonal projection in the
theory of harmonic fields 396
4. Construction of the operator A Z A. Proof of Kodaira and Gaffney theorems 398
* 5. Harmonic fields on compact manifolds 401
6. Concluding remarks 402
7. Some applications of harmonic fields in the investigation of compact Riemann
manifolds 403
CHAPTER XX
The general theory of differential operators (Hormander theory)
1. Introduction 406
2. The existence of a fundamental solution 408
3. Comparison of operators with constant coefficients 413
Methods of Hilbert spaces 2
it
18 Contents
4. Troves inequalities and their consequences 418
5. Operators of principal type with constant coefficients 424
6. A priori inequalities in the spaces H~s 429
7. Supplements and exercises 434
CHAPTER XXI
The ergodic theory
1. About the (quasi) ergodic hypothesis in statistical mechanics 441
2. Metric transitivity and ergodicity of a flow. V. Neumann ergodic theorem 442
3. Proof of the ergodic theorem 446
4. Direct proof of the mean ergodic theorem 446
5. The Yosida Kakutani ergodic theorem 448
6. Existence of invariant measure in compact dynamic systems 451
CHAPTER XXII
Theory of almost periodic functions and vectors. Representation of topological groups.
Spherical functions
1. Bochner definition of Bohr almost periodic functions. Almost periodic
vectors 453
2. Theory of almost periodic vectors 455
3. Theory of almost periodic vectors (continued). (Weyl Maak approximation
theorem) 458
4. Scalar valued almost periodic functions. Second fundamental theorem. The¬
orem of Bohr 459
5. Second fundamental theorem about almost periodic vectors 462
8. Almost periodic functions and group representations 463
7. Functions on homogeneous spaces. Spherical functions 465
CHAPTER XXIII
Hilbert space methods in complex analysis
Introduction 468
1. The inhomogeneous Cauchy Riemann equation and the Mittag Leffler problem
for functions of one complex variable 469
2. Some elementary facts from the theory of functions of several complex variables 471
3. Fundamental Existence Theorem and its applications to the Levi problem, the
Cousin problems, and the Oka, Cartan, Serre theorem 476
4. Proof of the fundamental theorem 483
5. Complex manifolds. Function theory on Stein manifolds 490
6. Holomorphic vector bundles. Approximation theorems of the Runge type for sec¬
tion of a holomorphic vector bundle 496
7. Cohomology groups of compact complex manifolds 503
8. Analyticity of generalized eigenvectors 506
Notes, exercises, complements 509
Conclusion. Historical and bibliographical remarks 513
Contents 19
Appendix I
1. Topology 520
2. Theory of the integral 522
Appendix II. Haar measure. Semi simple groups. Means of almost periodic vectors
1. Haar measure 529
2. Mean of an almost periodic vector 530
Appendix III. Proof of formula (11.8), Chapter XVII 535
Bibliography 537
List of symbols 544
Author index 546
Subject index 548
|
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| id | DE-604.BV003714788 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:04:17Z |
| institution | BVB |
| language | English Polish |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002364608 |
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| physical | 553 S. |
| publishDate | 1972 |
| publishDateSearch | 1972 |
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| publisher | PWN-Polish Scientific Publ. |
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| series2 | Monografie matematyczne |
| spelling | Maurin, Krzysztof 1923- Verfasser (DE-588)1014360323 aut Metody przestrzeni Hilberta Methods of Hilbert spaces Krzysztof Maurin 2. ed., rev. Warszawa PWN-Polish Scientific Publ. 1972 553 S. txt rdacontent n rdamedia nc rdacarrier Monografie matematyczne 45 Aus d. Poln. übers. Przestrzenie Hilberta jhpk Methode (DE-588)4038971-6 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Hilbert-Raum (DE-588)4159850-7 gnd rswk-swf Hilbert-Raum (DE-588)4159850-7 s Methode (DE-588)4038971-6 s DE-604 Funktionalanalysis (DE-588)4018916-8 s Monografie matematyczne 45 (DE-604)BV000003532 45 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002364608&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Maurin, Krzysztof 1923- Methods of Hilbert spaces Monografie matematyczne Przestrzenie Hilberta jhpk Methode (DE-588)4038971-6 gnd Funktionalanalysis (DE-588)4018916-8 gnd Hilbert-Raum (DE-588)4159850-7 gnd |
| subject_GND | (DE-588)4038971-6 (DE-588)4018916-8 (DE-588)4159850-7 |
| title | Methods of Hilbert spaces |
| title_alt | Metody przestrzeni Hilberta |
| title_auth | Methods of Hilbert spaces |
| title_exact_search | Methods of Hilbert spaces |
| title_full | Methods of Hilbert spaces Krzysztof Maurin |
| title_fullStr | Methods of Hilbert spaces Krzysztof Maurin |
| title_full_unstemmed | Methods of Hilbert spaces Krzysztof Maurin |
| title_short | Methods of Hilbert spaces |
| title_sort | methods of hilbert spaces |
| topic | Przestrzenie Hilberta jhpk Methode (DE-588)4038971-6 gnd Funktionalanalysis (DE-588)4018916-8 gnd Hilbert-Raum (DE-588)4159850-7 gnd |
| topic_facet | Przestrzenie Hilberta Methode Funktionalanalysis Hilbert-Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002364608&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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