Modern analysis and topology:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
New York u.a.
Springer
1995
|
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXVIII, 403 S. |
| ISBN: | 0387979867 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804124813376094208 |
|---|---|
| adam_text | CONTENTS
PREFACE vii
INTRODUCTION: TOPOLOGICAL BACKGROUND xvii
CHAPTER 1: METRIC SPACES 1
1.1 Metric and Pseudo Metric Spaces 1
Distance Functions, Spheres, Topology of Pseudo Metric Spaces, The
Ring C*(X), Real Hilbert Space, The Distance from a Point to a Set,
Partitions of Unity
1.2 Stone s Theorem 6
Refinements, Star Refinements and A Refinements, Full Normality,
Paracompactness, Shrinkable Coverings, Stone s Theorem
1.3 The Metrization Problem 13
Functions That Can Distinguish Points from Sets, o Local Finiteness,
Urysohn s Metrization Theorem, The Nagata Smimov Metrization
Theorem, Local Starrings, Arhangel skil s Metrization Theorem
1.4 Topology of Metric Spaces 20
Complete Normality and Perfect Normality, First and Second Count¬
able Spaces, Separable Spaces, The Diameter of a Set, The Lebesgue
Number, Precompact Spaces, Countably Compact and Sequentially
Compact Spaces
1.5 Uniform Continuity and Uniform Convergence 25
Uniform Continuity, Uniform Homeomorphisms and Isomorphisms,
Isometric Functions, Uniform Convergence
1.6 Completeness 28
Convergence and Clustering of Sequences, Cauchy Sequences and
Cofinally Cauchy, Sequences, Complete and Cofinally Complete
Spaces, The Lebesgue Property, Borel Compactness, Regularly
Bounded Metric Spaces
1.7 Completions 38
The Completion of a Metric Space, Uniformly Continuous Extensions
CHAPTER 2: UNIFORMITIES 43
2.1 Covering Uniformities 43
Uniform Spaces, Normal Sequences of Coverings, Bases and Subbases
for Uniformities, Normal Coverings, Uniform Topology
x Contents
2.2 Uniform Continuity 48
Uniform Continuity, Uniform Homeomorphisms, Pseudo Metrics
Determined by Normal Sequences
2.3 Uniformizability and Complete Regularity 52
Uniformizable Spaces, The Equivalence of Uniformizability and
Complete Regularity, Regularly Open Sets and Coverings, Open and
Closed Bases of Uniformities, Regularly Open Bases of Uniformities,
Universal or Fine Uniformities
2.4 Normal Coverings 56
The Unique Uniformity of a Compact Hausdorff Space, Tukey s
Characterization of Normal Spaces, Star Finite Coverings, Precise
Refinements, Some Results of K. Morita, Some Corrections of Tukey s
Theorems by Morita
CHAPTER 3: TRANSFINITE SEQUENCES 62
3.1 Background 62
3.2 Transfinite Sequences in Uniform Spaces 63
Cauchy and Cofinally Cauchy Transfinite Sequences, A Character¬
ization of Paracompactness in Terms of Transfinite Sequences,
Shirota s e Uniformity, Some Characterizations of the Lindelof
Property in Terms of Transfinite Sequences, The P Uniformity, A
Characterization of Compactness in Terms of Transfinite Sequences
3.3 Transfinite Sequences and Topologies 75
Characterizations of Open and Closed Sets in Terms of Transfinite
Sequences, A Characterization of the Hausdorff Property in Terms of
Transfinite Sequences, Cluster Classes and the Characterization of
Topologies, A Characterization of Continuity in Terms of Transfinite
Sequences
CHAPTER 4: COMPLETENESS, COFINAL COMPLETENESS AND
UNIFORM PARACOMPACTNESS 83
4.1 Introduction 83
4.2 Nets 84
Convergence and Clustering of Nets, Characterizations of Open and
Closed Sets in Terms of Nets, A Characterization of the Hausdorff
Property in Terms of Nets, Subnets, A Characterization of
Contents xi
Compactness in Terms of Nets, A Characterization of Continuity in
Terms of Nets, Convergence Classes and the Characterization of
Topologies, Universal Nets, Characterizations of Paracompactness, the
Lindelof Property and Compactness in Terms of Nets
4.3 Completeness, Cofinal Completeness and Uniform Paracompactness 92
Cauchy and Cofinally Cauchy Nets, Completeness and Cofinal Com¬
pleteness, The Lebesgue Property, Precompactness, Uniform Para¬
compactness
4.4 The Completion of a Uniform Space 97
Fundamental Nets, Completeness in Terms of Fundamental Nets, The
Construction of the Completion with Fundamental Nets, The Unique¬
ness of the Completion
4.5 The Cofinal Completion or Uniform Paracompactification 103
The Topological Completion, Preparacompactness, Countable Bound
edness and the Lindelof Property, A Necessary and Sufficient Con¬
dition for a Uniform Space to Have a Paracompact Completion, A
Necessary and Sufficient Condition for a Uniform Space to Have a
Lindelof Completion, The Existence of the Cofinal Completion, A
Characterization of Preparacompactness
Chapter 5: FUNDAMENTAL CONSTRUCTIONS 110
5.1 Introduction 110
5.2 Limit Uniformities 111
Infimum and Supremum Topologies, Infimum and Supremum Uniform¬
ities, Projective and Inductive Limit Topologies, Projective and Induc¬
tive Limit Uniformities
5.3 Subspaces, Sums, Products and Quotients 114
Uniform Product Spaces, Uniform Subspaces, Quotient Uniform
Spaces, The Uniform Sum
5.4 Hyperspaces 119
The Hyperspace of a Uniform Space, Supercompleteness, Burdick s
Characterization of Supercompleteness, Other Characterizations of
Supercompleteness, Supercompleteness and Cofinal Completeness,
Paracompactness and Supercompleteness
xii Contents
5.5 Inverse Limits and Spectra 126
Inverse Limit Sequences, Inverse Limit Systems, Inverse Limit
Systems of Uniform Spaces, Morita s Weak Completion, The Spectrum
of Weakly Complete Uniform Spaces, Morita s and Pasynkov s
Characterizations of Closed Subsets of Products of Metric Spaces
5.6 The Locally Fine Coreflection 133
Uniformly Locally Uniform Coverings, Locally Fine Uniform Spaces,
The Derivative of a Uniformity, Partially Cauchy Nets, Injective
Uniform Spaces, Subfine Uniform Spaces, The Subfine Coreflection
5.7 Categories and Functors 146
Concrete Categories, Objects, Morphisms, Covariant Functors, Isomor¬
phisms, Monomorphisms, Duality, Subcategories, Reflection, Core¬
flection
CHAPTER 6: PARACOMPACTIFICATIONS 156
6.1 Introduction 156
Some Problems of K. Morita and H. Tamano, Topological Completion,
Paracompactifications, Compactifications, Samuel Compactifications,
The Stone Cech Compactification, Uniform Paracompactifications,
Tamano s Paracompactification Problem
6.2 Compactifications 159
Extensions of Open Sets, Extensions of Coverings, The Extent of a
Covering, Stable Coverings, Star Finite Partitions of Unity
6.3 Tamano s Completeness Theorem 171
The Radical of a Uniform Space, Tamano s Completeness Theorem,
Necessary and Sufficient Conditions for Topological Completeness
6.4 Points at Infinity and Tamano s Theorem 178
Points and Sets at Infinity, Some Characterizations of Paracompactness
by Tamano, Tamano s Theorem
6.5 Paracompactifications 182
Completions of Uniform Spaces as Subsets of pX, A Solution of
Tamano s Paracompactification Problem, The Tamano Morita Para¬
compactification, Characterizations of Paracompactness, the Lindelof
Property and Compactness in Terms of Supercompleteness, Another
Necessary and Sufficient Condition for a Uniform Space to Have a
Paracompact Completion, Another Necessary and Sufficient Condition
Contents xiii
for a Uniform Space to Have a Lindelof Completion, The Definition
and Existence of the Supercompletion
6.6 The Spectrum of PX 192
The Spectrum of PX, The Spectrum of uX, Morita s Weak Completion
6.7 The Tamano Morita Paracompactification 197
M spaces, Perfect and Quasi perfect Mappings, The Topological Com¬
pletion of an M space, The Tamano Morita Paracompactification of an
M space
CHAPTER 7: REALCOMPACTTFICATIONS 202
7.1 Introduction 202
Another Characterization of PX, Q spaces, CZ maximal Families
7.2 Realcompact Spaces 203
Realcompact Spaces, The Hewitt Realcompactification, Character¬
izations of Realcompactness, Properties of Realcompact Spaces,
Pseudo metric Uniformities, The c and c* Uniformities
7.3 Realcompactifications 210
Realcompactifications, The Equivalence of viX and eX, The Unique¬
ness of the Hewitt Realcompactification, Characterizations of i)X,
Properties of vX, Hereditary Realcompactness
7.4 Realcompact Spaces and Lindelof Spaces 217
Tamano s Characterization of Realcompact Spaces, A Necessary and
Sufficient Condition for the Realcompactification to be Lindelof,
Tamano s Characterization of Lindelof Spaces
7.5 Shirota s Theorem 221
Measurable Cardinals, {0,1} Measures, The Relationship of Non Zero
{0,1} Measures and CZ maximal Families, A Necessary and Sufficient
Condition for Discrete Spaces to be Realcompact, Closed Classes of
Cardinals, Shirota s Theorem
CHAPTER 8: MEASURE AND INTEGRATION 229
8.1 Introduction 229
Riemann Integration, Lebesgue Integration, Measures, Invariant
Integrals
xiv Contents
8.2 Measure Rings and Algebras 230
Rings, Algebras, o Rings, o Algebras, Borel Sets, Baire Sets,
Measures, Measure Rings, Measurable Sets, Measure Algebras,
Measure Spaces, Complete Measures, The Completion of a Measure,
Borel Measures, Lebesgue Measure, Baire Measures, The Lebesgue
Ring, Lebesgue Measurable Sets, Finite Measures, Infinite Measures
8.3 Properties of Measures 235
Monotone Collections, Continuous from Below, Continuous from
Above
8.4 Outer Measures 238
Hereditary Collections, Outer Measures, Extensions of Measures,
H* Measurability
8.5 Measurable Functions 243
Measurable Spaces, Measurable Sets, Measurable Functions, Borel
Functions, Limits Superior, Limits Inferior, Point wise Limits of
Functions, Simple Functions, Simple Measurable Functions
8.6 The Lebesgue Integral 249
Development of the Lebesgue Integral
8.7 Negligible Sets 256
Negligible Sets, Almost Everywhere, Complete Measures, Completion
of a Measure
8.8 Linear Functional and Integrals 257
Linear Functional, Positive Linear Functional, Lower Semi con¬
tinuous, Upper Semi continuous, Outer Regularity, Inner Regularity,
Regular Measures, Almost Regular Measures, The Riesz Represen¬
tation Theorem
CHAPTER 9: HAAR MEASURE IN UNIFORM SPACES 264
9.1 Introduction 264
Isogeneous Uniform Spaces, Isomorphisms, Homogeneous Spaces,
Translations, Rotations, Reflections, Haar Integral, Haar Measure
9.2 Haar Integrals and Measures 267
Development of the Haar integral on Locally Compact Isogeneous
Uniform Spaces
Contents xv
9.3 Topological Groups and Uniqueness of Haar Measures 271
Topological Groups, Abelian Topological Groups, Open at 0, Right
Uniformity, Left Uniformity, Right Coset, Left Coset, Quotient of a
Topological Group, A Necessary and Sufficient Condition for a
Locally Compact Space to Have a Topological Group Structure
CHAPTER 10: UNIFORM MEASURES 284
10.1 Introduction 284
Uniform Measures, The Congruence Axiom, Loomis Contents
10.2 Prerings and Loomis Contents 285
Prerings, Hereditary Open Prerings, Loomis Contents, Uniformly
Separated, Left Continuity, Invariant Loomis Contents, Zero boundary
Sets
10.3 The Haar Functions 292
The Haar Covering Function, The Haar Function, Extension of Loomis
Contents to Finitely Additive Measures
10.4 Invariance and Uniqueness of Loomis Contents and Haar Measures 299
Invariance with Respect to a Uniform Covering, Invariance on Com¬
pact Spheres, Development of Loomis Contents on Suitably Restric¬
ted Uniform Spaces.
10.5 Local Compactness and Uniform Measures 304
Almost Uniform Measures, Uniform Measures, Jordan Contents,
Monotone Sequences of Sets, Monotone Classes, Development of
Uniform Measures on Suitably Restricted Uniform Spaces
CHAPTER 11: SPACES OF FUNCTIONS 317
11.1 {/ spaces 317
Conjugate Exponents, Lp norm, The Essential Supremum, Essentially
Bounded, Minkowski s Inequality iolder s Inequality, The Supremum
Norm, The Completion of C*(X) with Respect to the Z/ norm
11.2 The Space L2( i) and Hilbert Spaces 326
Square Integrable Functions, Inner Product, Schwarz Inequality,
Hilbert Space, Orthogonality, Orthogonal Projections, Linear Combin¬
ations, Linear Independence, Span, Basis of a Vector Space,
xvi Contents
Orthonormal Sets, Orthonormal Bases, Bessel s Inequality, Riesz
Fischer Theorem, Hilbert Space Isomorphism
11.3 The Space Lpdi) and Banach Spaces 340
Normed Linear Space, Banach Space, Linear Operators, Kernel of a
Linear Operator, Bounded Linear Operators, Dual Spaces, Hahn
Banach Theorem, Second Dual Space, Baire s Category Theorem,
Nowhere Dense Sets, Open Mapping Theorem, Closed Graph
Theorem, Uniform Boundedness Principle, Banach Steinhaus Theorem
11.4 Uniform Function Spaces 355
Uniformity of Pointwise Convergence, Uniformity of Uniform
Convergence, Joint Continuity, Uniformity of Uniform Convergence
on Compacta, Topology of Compact Convergence, Compact Open
Topology, Joint Continuity on Compacta, Ascoli Theorem, Equicon
tinuity
CHAPTER 12: UNIFORM DIFFERENTIATION 370
12.1 Complex Measures 370
Complex Measure, Total Variation, Absolute Continuity,
Concentration of a Measure on a Subset, Orthogonality of Measures
12.2 The Radon Nikodym Derivative 373
Radon Nikodym Derivative and its Applications
12.3 Decompositions of Measures and Complex Integration 380
Polar Decomposition, Lebesgue Decomposition, Complex Integration
12.4 The Riesz Representation Theorem 386
Regular and Almost Regular Complex Measures, The Riesz Repre¬
sentation Theorem
12.5 Uniform Derivatives of Measures 389
Differentiation of a Measure at a Point, Differentiable Measures,
L differentiable Measures, Uniformly Differentiable Measures,
Fubini s Theorem
INDEX 394
|
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| indexdate | 2024-07-09T17:51:38Z |
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| series2 | Universitext |
| spelling | Howes, Norman R. Verfasser aut Modern analysis and topology Norman R. Howes New York u.a. Springer 1995 XXVIII, 403 S. txt rdacontent n rdamedia nc rdacarrier Universitext Mathematical analysis Topology Analysis (DE-588)4001865-9 gnd rswk-swf Uniformer Raum (DE-588)4137585-3 gnd rswk-swf Integration Mathematik (DE-588)4072852-3 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Topologie (DE-588)4060425-1 s Uniformer Raum (DE-588)4137585-3 s Integration Mathematik (DE-588)4072852-3 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006916507&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Howes, Norman R. Modern analysis and topology Mathematical analysis Topology Analysis (DE-588)4001865-9 gnd Uniformer Raum (DE-588)4137585-3 gnd Integration Mathematik (DE-588)4072852-3 gnd Topologie (DE-588)4060425-1 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4137585-3 (DE-588)4072852-3 (DE-588)4060425-1 |
| title | Modern analysis and topology |
| title_auth | Modern analysis and topology |
| title_exact_search | Modern analysis and topology |
| title_full | Modern analysis and topology Norman R. Howes |
| title_fullStr | Modern analysis and topology Norman R. Howes |
| title_full_unstemmed | Modern analysis and topology Norman R. Howes |
| title_short | Modern analysis and topology |
| title_sort | modern analysis and topology |
| topic | Mathematical analysis Topology Analysis (DE-588)4001865-9 gnd Uniformer Raum (DE-588)4137585-3 gnd Integration Mathematik (DE-588)4072852-3 gnd Topologie (DE-588)4060425-1 gnd |
| topic_facet | Mathematical analysis Topology Analysis Uniformer Raum Integration Mathematik Topologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006916507&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT howesnormanr modernanalysisandtopology |