Some modern mathematics for physicists and other outsiders: an introduction to algebra, topology, and functional analysis 1 Algebra, topology, and measure theory
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Pergamon Press
1975
|
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XLVIII, 378 S. graph. Darst. |
| ISBN: | 0080180965 0080180973 |
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MARC
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| 100 | 1 | |a Roman, Paul |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Some modern mathematics for physicists and other outsiders |b an introduction to algebra, topology, and functional analysis |n 1 |p Algebra, topology, and measure theory |c Paul Roman |
| 264 | 1 | |a New York [u.a.] |b Pergamon Press |c 1975 | |
| 300 | |a XLVIII, 378 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804136016898949120 |
|---|---|
| adam_text | Contents of Volume 1
Contents of Volume 2 x
Preface xiii
Organization of the Book xvii
Acknowledgments xxi
Introduction xxiii
PART ONE: THE RAW MATERIALS OF MATHEMATICS 1
Chapter 1 Sets 3
1.1 Operations with Sets 6
1.2 Relations in Sets 13
1.2a. Equivalence relations 15
1.2b. Order relations 20
Chapter 2 Maps 26
2.1 Composite Functions and Inverses 32
2.2 Equivalence Relations and Maps 38
2.3 Ordered Sets and Maps 42
2.4 Cardinal Numbers 43
2.5 Sequences and Families 47
vii
viii Contents
PART TWO: THE BASIC STRUCTURES OF MATHEMATICS 53
IIA: Algebraic Structures 55
Chapter 3 Algebraic Composition Laws and Systems 57
3.1 Morphisms of Algebraic Systems 62
Chapter 4 Survey of Special Algebraic Systems 68
4.1 Groups 70
4.1a. Transformation groups; G spaces; orbits 79
4.1b. Conjugate classes; cosets 87
4.1c. Normal subgroups; quotient groups; isomorphism
theorems 91
4.2 Rings and Fields 102
4.2a. Ideals; quotient rings; isomorphism theorems 115
4.3 Linear Spaces 120
4.3a. Linear independence, bases and dimension 128
4.3b. Morphisms (linear transformations); quotient spaces 137
4.4 Linear Algebras 149
4.4a. Morphisms of algebras; quotient algebras 159
4.5 Nonassociative Algebras 167
4.5a. Lie algebras 168
4.5b. Some other nonassociative algebras 182
IIB: Topological Structures 185
Chapter 5 Topological Spaces 187
5.1 Examples; Metric Spaces 188
5.2 General Structure of Topological Spaces 198
5.3 Neighborhoods; Special Points; Closed Sets 203
5.3a. Interior, closure, boundary 208
5.4 Convergence 211
5.5 Continuity 216
5.6 Homeomorphism and Isometry 220
5.6a. Quotient topology; homeomorphism theorem 229
Chapter 6 Topological Spaces with Special Properties 236
6.1 Connected Spaces 236
6.1a. Path connectivity; homotopy 242
6.2 Separable Spaces 250
Contents ix
6.3 Compact Spaces 254
6.3a. Compactification 266
6.4 Complete Metric Spaces 270
6.4a. Completion 275
6.4b. Contraction mappings 280
IIC: Measure Structures 291
Chapter 7 Measure Spaces 293
7.1 Measurable Spaces 294
7.2 Measure and Measure Spaces 305
7.2a. General properties of measures 310
7.2b. Lebesgue measure 314
7.2c. Lebesgue Stieltjes measures 320
7.2d. Signed and complex measure 325
Chapter 8 Theory of Integration 328
8.1 Measurable Functions 329
8.2 Definition of the Integral 338
8.3 General Properties of the Integral 353
8.4 Comments on Lebesgue and Lebesgue Stieltjes Integrals 365
8.5 The Radon Nikodym Theorem 371
APPENDICES xxvii
Appendix I Some Inequalities xxix
Appendix m Annotated Reading List xxx
Appendix IV Frequently Used Symbols xxxvi
Index xliii
|
| adam_txt |
Contents of Volume 1
Contents of Volume 2 x
Preface xiii
Organization of the Book xvii
Acknowledgments xxi
Introduction xxiii
PART ONE: THE RAW MATERIALS OF MATHEMATICS 1
Chapter 1 Sets 3
1.1 Operations with Sets 6
1.2 Relations in Sets 13
1.2a. Equivalence relations 15
1.2b. Order relations 20
Chapter 2 Maps 26
2.1 Composite Functions and Inverses 32
2.2 Equivalence Relations and Maps 38
2.3 Ordered Sets and Maps 42
2.4 Cardinal Numbers 43
2.5 Sequences and Families 47
vii
viii Contents
PART TWO: THE BASIC STRUCTURES OF MATHEMATICS 53
IIA: Algebraic Structures 55
Chapter 3 Algebraic Composition Laws and Systems 57
3.1 Morphisms of Algebraic Systems 62
Chapter 4 Survey of Special Algebraic Systems 68
4.1 Groups 70
4.1a. Transformation groups; G spaces; orbits 79
4.1b. Conjugate classes; cosets 87
4.1c. Normal subgroups; quotient groups; isomorphism
theorems 91
4.2 Rings and Fields 102
4.2a. Ideals; quotient rings; isomorphism theorems 115
4.3 Linear Spaces 120
4.3a. Linear independence, bases and dimension 128
4.3b. Morphisms (linear transformations); quotient spaces 137
4.4 Linear Algebras 149
4.4a. Morphisms of algebras; quotient algebras 159
4.5 Nonassociative Algebras 167
4.5a. Lie algebras 168
4.5b. Some other nonassociative algebras 182
IIB: Topological Structures 185
Chapter 5 Topological Spaces 187
5.1 Examples; Metric Spaces 188
5.2 General Structure of Topological Spaces 198
5.3 Neighborhoods; Special Points; Closed Sets 203
5.3a. Interior, closure, boundary 208
5.4 Convergence 211
5.5 Continuity 216
5.6 Homeomorphism and Isometry 220
5.6a. Quotient topology; homeomorphism theorem 229
Chapter 6 Topological Spaces with Special Properties 236
6.1 Connected Spaces 236
6.1a. Path connectivity; homotopy 242
6.2 Separable Spaces 250
Contents ix
6.3 Compact Spaces 254
6.3a. Compactification 266
6.4 Complete Metric Spaces 270
6.4a. Completion 275
6.4b. Contraction mappings 280
IIC: Measure Structures 291
Chapter 7 Measure Spaces 293
7.1 Measurable Spaces 294
7.2 Measure and Measure Spaces 305
7.2a. General properties of measures 310
7.2b. Lebesgue measure 314
7.2c. Lebesgue Stieltjes measures 320
7.2d. Signed and complex measure 325
Chapter 8 Theory of Integration 328
8.1 Measurable Functions 329
8.2 Definition of the Integral 338
8.3 General Properties of the Integral 353
8.4 Comments on Lebesgue and Lebesgue Stieltjes Integrals 365
8.5 The Radon Nikodym Theorem 371
APPENDICES xxvii
Appendix I Some Inequalities xxix
Appendix m Annotated Reading List xxx
Appendix IV Frequently Used Symbols xxxvi
Index xliii |
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| isbn | 0080180965 0080180973 |
| language | English |
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| physical | XLVIII, 378 S. graph. Darst. |
| publishDate | 1975 |
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| publisher | Pergamon Press |
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| spelling | Roman, Paul Verfasser aut Some modern mathematics for physicists and other outsiders an introduction to algebra, topology, and functional analysis 1 Algebra, topology, and measure theory Paul Roman New York [u.a.] Pergamon Press 1975 XLVIII, 378 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier (DE-604)BV009346612 1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015256029&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Roman, Paul Some modern mathematics for physicists and other outsiders an introduction to algebra, topology, and functional analysis |
| title | Some modern mathematics for physicists and other outsiders an introduction to algebra, topology, and functional analysis |
| title_auth | Some modern mathematics for physicists and other outsiders an introduction to algebra, topology, and functional analysis |
| title_exact_search | Some modern mathematics for physicists and other outsiders an introduction to algebra, topology, and functional analysis |
| title_exact_search_txtP | Some modern mathematics for physicists and other outsiders an introduction to algebra, topology, and functional analysis |
| title_full | Some modern mathematics for physicists and other outsiders an introduction to algebra, topology, and functional analysis 1 Algebra, topology, and measure theory Paul Roman |
| title_fullStr | Some modern mathematics for physicists and other outsiders an introduction to algebra, topology, and functional analysis 1 Algebra, topology, and measure theory Paul Roman |
| title_full_unstemmed | Some modern mathematics for physicists and other outsiders an introduction to algebra, topology, and functional analysis 1 Algebra, topology, and measure theory Paul Roman |
| title_short | Some modern mathematics for physicists and other outsiders |
| title_sort | some modern mathematics for physicists and other outsiders an introduction to algebra topology and functional analysis algebra topology and measure theory |
| title_sub | an introduction to algebra, topology, and functional analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015256029&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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