Theory of operators:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Consultants Bureau
1991
|
| Edition: | 2. ed. |
| Series: | Contemporary Soviet mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | XI, 396 S. |
| ISBN: | 0306110288 |
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MARC
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| 245 | 1 | 0 | |a Theory of operators |c V. A. Sadovnichiǐ |
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| _version_ | 1804120872020082688 |
|---|---|
| adam_text | THEORY OF OPERATORS
V A Sadovnichii
Moscow State University
Moscow, USSR
Translated from Russian by
Roger Cooke
CONSULTANTS BUREAU • NEW YORK AND LONDON
Contents
Chapter 1 Metric and Topological Spaces 1
1 Elementary Concepts of Set Theory 1
1 1 Elementary Properties of Sets Mappings
Cartesian Product of Sets 1
1 2 The Cardinality of a Set 7
1 3 Partial Ordering Ordering 10
1 4 Comparison of Cardinalities 11
2 Metric Spaces 13
2 1 Definition of a Metric Space Examples 13
2 2 Open and Closed Sets 18
2 3 Everywhere-Dense and Perfect Sets 21
2 4 Convergence Continuous Mappings 23
2 5 Compactness 25
26A Basis for the Topology of a Space 27
Exercises 30
3 Properties of Metric Spaces 31
3 1 Completion of Metric Spaces 33
3 2 Basic Theorems on Complete Metric Spaces 35
3 3 Compactness in Metric Spaces, e-Nets 43
Exercises ; 45
4 Topological Spaces 46
4 1 Definition of a Topological Space
Hausdorff spaces Examples 46
vii
viii Contents
42A Remark on Topological Spaces 50
Exercises 52
5 Properties of Topological Spaces 53
5 1 Regular, Completely Regular, and Normal Spaces 53
5 2 Regular Spaces with a Countable Basis
Tikhonov s Theorem 55
5 3 Compact Hausdorff and Normal Spaces 56
5 4 Metric and Topological Spaces 57
5 5 Cartesian Products of Topological Spaces 58
5 6 The Stone-Weierstrass Theorem 60
Exercises 62
Chapter 2 Vector Spaces 65
1 Topological Vector Spaces 65
1 1 Group, Ring, Field, and Vector Space 65
1 2 Linear Transformations
The Space of Transformations 71
1 3 Banach Spaces 72
1 4 Convex Sets, the Minkowski Functional,
Seminorms 73
1 5 Topological Vector Spaces Kolmogorov s Theorem 79
1 6 Countably Normed Spaces - 85
Exercises 87
2 Bounded Linear Transformations on Banach and
F-Spaces Basic Principles of Functional Analysis 88
2 1 Bounded Linear Transformations on Banach
Spaces The Banach Space of Transformations
The Concept of an F-space 89
2 2 The Uniform Boundedness Principle 93
2 3 The Bounded Inverse Theorem
The Open Mapping Principle 99
2 4 Extension of Transformations and Functionals
The Hahn-Banach Extension Theorem 103
2 5 Different Topologies Different Types of
Convergence The General Forms of
Functionals in Particular Spaces 109
2 6 Compact Sets, Weak-Star Compactness 120
Exercises 124
Contents be
Chapter 3 Measure Theory Measurable
Functions and Integration 127
1 Measure Theory 127
2 Measurable Functions 143
3 The Lebesgue Integral 147
3 1 Definition of the Lebesgue Integral 149
3 2 Properties of the Lebesgue Integral 151
3 3 Passage to the Limit under the Integral Sign 158
3 4 The Connection between the Riemann and
Lebesgue Integrals 162
3 5 The Space 1/ 164
4 Absolutely Continuous Set Functions
The Radon-Nikodym Theorem 168
4 1 Absolutely Continuous Set Functions 168
4 2 The Radon-Nikodym Theorem 169
5 Product Measures Fubini s Theorem 173
5 1 Product Measures 173
5 2 Fubini s Theorem 178
Exercises 179
Chapter 4 The Geometry of Hilbert Space
The Spectral Theory of Operators 181
1 Hilbert Spaces 181
1 1 The Geometry of a Hilbert Space 182
1 2 Bases of a Hilbert Space 187
1 3 The Dimension of a Hilbert Space 191
1 4 Orthogonal Expansions in Hilbert Space 193
1 5 Biorthogonal Sequences 194
1 6 The Matrix Representation of a Bounded
Linear Operator on H 202
Exercises 208
2 Spectral Theorems 210
2 1 The Adjoint Operator 210
2 2 The Concept of a
Completely Continuous Operator 212
2 3 The Absolute Norm of an Operator 213
2 4 The Fredholm Alternative 217
2 5 Projections 222
x Contents
2 6 The Spectrum of an Operator 225
2 7 Symmetric Operators Properties of the
Quadratic Form of an Operator 228
2 8 The Square Root of a Symmetric Operator 231
2 9 The Spectral Theorem for a Symmetric
Operator on n-Dimensional Space 232
2 10 Completely Continuous Operators
The Spectral Theorem 236
2 11 The Spectral Theorem for a Bounded
Symmetric Operator 238
2 12 The Spectral Theorem for a Unitary Operator 243
2 13 Unbounded Operators 251
2 14 The Spectrum of a Symmetric Bounded Operator 266
2 15 The Spectrum and Resolvent of
Unbounded Operators 271
Exercises 275
3 Operator Equations Analytic Functions and Operators 277
3 1 Analytic Properties of the Resolvent 277
3 2 Keldysh s Theorem 287
3 3 Root Vectors and Root Subspaces of
Nonself-adjoint Operators 290
3 4 Differential Operators 299
Chapter 5 The Trace of an Operator 307
1 The Trace Theorem for an Operator in
n-Dimensional Space 307
2 Nuclear Operators The Trace Theorem 308
2 1 The Trace Theorem for a Positive
Nuclear Operator 308
2 2 Properties of the s-Numbers of
Completely Continuous Operators 313
2 3 Estimates for the Eigenvalues of a Completely
Continuous Operator 321
2 4 Estimates for the s-Numbers of Products and
Sums of Completely Continuous Linear Operators 328
2 5 The Trace Theorem for a Nuclear Operator 330
3 Regularized Sums of Zeros of a Class of Entire Functions
The Trace of a Differential Operator 338
3 1 Functions of Class K 338
Contents xi
3 2 The Zeta-Function 340
3 3 Regularized Sums of Zeros of a
Function of Class K 343
Exercises 347
4 The Trace of a Discrete Operator 349
Chapter 6 Distributions The Fourier Transform 361
1 Distributions 361
1 1 The Concept of a Distribution 361
1 2 Basic Properties of Distributions 367
1 3 Differential Equations in Distributions 372
1 4 The Tensor Product and Convolution of
Distributions 375
2 The Fourier Transform 378
2 1 The Fourier Transform of Functions of the
Space L1 378
2 2 The Fourier Transform of Functions in the
Space L2 381
2 3 The Fourier Transform of a Distribution 383
References 387
Index 389
|
| any_adam_object | 1 |
| author | Sadovničij, Viktor Antonovič 1939- |
| author_GND | (DE-588)1017832552 |
| author_facet | Sadovničij, Viktor Antonovič 1939- |
| author_role | aut |
| author_sort | Sadovničij, Viktor Antonovič 1939- |
| author_variant | v a s va vas |
| building | Verbundindex |
| bvnumber | BV006598538 |
| classification_rvk | SK 620 |
| ctrlnum | (OCoLC)246785515 (DE-599)BVBBV006598538 |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV006598538 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:48:59Z |
| institution | BVB |
| isbn | 0306110288 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-004213891 |
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| physical | XI, 396 S. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| publisher | Consultants Bureau |
| record_format | marc |
| series2 | Contemporary Soviet mathematics |
| spelling | Sadovničij, Viktor Antonovič 1939- Verfasser (DE-588)1017832552 aut Theory of operators V. A. Sadovnichiǐ 2. ed. New York u.a. Consultants Bureau 1991 XI, 396 S. txt rdacontent n rdamedia nc rdacarrier Contemporary Soviet mathematics Aus dem Russ. übers. Operatortheorie (DE-588)4075665-8 gnd rswk-swf Operatortheorie (DE-588)4075665-8 s DE-604 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004213891&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sadovničij, Viktor Antonovič 1939- Theory of operators Operatortheorie (DE-588)4075665-8 gnd |
| subject_GND | (DE-588)4075665-8 |
| title | Theory of operators |
| title_auth | Theory of operators |
| title_exact_search | Theory of operators |
| title_full | Theory of operators V. A. Sadovnichiǐ |
| title_fullStr | Theory of operators V. A. Sadovnichiǐ |
| title_full_unstemmed | Theory of operators V. A. Sadovnichiǐ |
| title_short | Theory of operators |
| title_sort | theory of operators |
| topic | Operatortheorie (DE-588)4075665-8 gnd |
| topic_facet | Operatortheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004213891&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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