Introduction to the analysis of metric spaces:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
1987
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Australian Mathematical Society: Australian Mathematical Society lecture series
3 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 257 S. graph. Darst. |
| ISBN: | 0521350514 0521359287 |
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| 100 | 1 | |a Giles, John R. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to the analysis of metric spaces |c J. R. Giles |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1987 | |
| 300 | |a XIV, 257 S. |b graph. Darst. | ||
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| 490 | 1 | |a Australian Mathematical Society: Australian Mathematical Society lecture series |v 3 | |
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| 650 | 4 | |a Espaces linéaires normés | |
| 650 | 4 | |a Espaces métriques | |
| 650 | 4 | |a Functional analysis | |
| 650 | 4 | |a Metric spaces | |
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Datensatz im Suchindex
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| adam_text | CONTENTS page
PREFACE viii
Chapter I. METRIC SPACES AND NORMED LINEAR SPACES 1
1. DEFINITIONS AND EXAMPLES 1
Metric spaces, normed linear spaces; metrics
generated by a norm; co ordinate, sequence and
function spaces; semi normed linear spaces;
Exercises.
2. BALLS AND BOUNDEDNESS 21
Balls and spheres in metric spaces and normed
linear spaces, relating norms and balls;
boundedness, diameter; distances between sets;
Exercises.
Chapter II. LIMIT PROCESSES 36
3. CONVERGENCE AND COMPLETENESS 36
Convergence of sequences, characterisation in
finite dimensional normed linear spaces,
uniform convergence; equivalent metrics and
norms; Cauchy sequences, completeness;
convergence of series; Exercises.
4. CLUSTER POINTS AND CLOSURE 66
Cluster points, closed sets; relating closed
to complete; closure, density, separability;
the boundary of a set; Exercises.
5. APPLICATION: BANACH S FIXED POINT THEOREM 91
Fixed points, Banach s Fixed Point Theorem.
5.7 Application in real analysis 93
5.8 Application in linear algebra 96
5.9 Application in the theory of differential
equations 100
Picard s Theorem
5.10 Application in the theory of integral equations 103
Fredholm integral equations, Volterra
integral equations
Exercises.
vi Contents
Chapter III. CONTINUITY 114
6. CONTINUITY IN METRIC SPACES 114 j.
Local continuity, characterisation of continuity
by sequences, algebra of continuous mappings;
global continuity characterised by inverse
images; isometrics, homeomorphisms; uniform
continuity; Exercises.
7. CONTINUOUS LINEAR MAPPINGS 138 v
Characterisation of continuity of linear mappings,
linear mappings on finite dimensional normed linear
spaces, continuity of linear functionals;
topological isomorphisms, isometric isomorphisms;
Exercises.
Chapter IV. COMPACTNESS 160
8. SEQUENTIAL COMPACTNESS IN METRIC SPACES 161
Properties of compact sets; characterisation
in finite dimensional normed linear spaces, ;
Riesz Theorem; application in approximation
theory; alternative forms of compactness,
total boundedness, ball cover compactness;
separability; Exercises.
9. CONTINUOUS FUNCTIONS ON COMPACT METRIC SPACES 183
Heine s Theorem, Dini s Theorem
9.9 The structure of the real Banach space
(C[a,b],ll IIJ 187
The Weierstrass Approximation Theorem
9.10 The structure of the Banach space (C(X),II IIJ
where (X,d) is a compact metric space . 194
9.11 Compactness in (C(X), II •IIJ 200
equicontinuity, The Ascoli Arzela Theorem,
Peano s Theorem
Exercises.
Contents vi
Chapter V. THE METRIC TOPOLOGY 213
10. THE TOPOLOGICAL ANALYSIS OF METRIC SPACES 214
Open sets and their properties, base for a
topology; equivalent metrics; relation to
closed sets; the interior of a set; the
characterisation of continuous mappings by
inverse images; topological compactness;
separability, the normal topological structure;
Exercises.
APPENDICES 235
Appendix 1. The real analysis background 235
Appendix 2. The set theory background 240
Appendix 3. The linear algebra background 246
INDEX TO NOTATION 251
INDEX 253
|
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| discipline | Mathematik |
| edition | 1. publ. |
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| indexdate | 2024-07-09T15:39:44Z |
| institution | BVB |
| isbn | 0521350514 0521359287 |
| language | English |
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| physical | XIV, 257 S. graph. Darst. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | Cambridge Univ. Press |
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| series2 | Australian Mathematical Society: Australian Mathematical Society lecture series |
| spelling | Giles, John R. Verfasser aut Introduction to the analysis of metric spaces J. R. Giles 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1987 XIV, 257 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Australian Mathematical Society: Australian Mathematical Society lecture series 3 Analyse fonctionnelle Espaces linéaires normés Espaces métriques Functional analysis Metric spaces Normed linear spaces Analysis (DE-588)4001865-9 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Normierter Raum (DE-588)4127735-1 gnd rswk-swf Normierter Raum (DE-588)4127735-1 s Analysis (DE-588)4001865-9 s DE-604 Metrischer Raum (DE-588)4169745-5 s Australian Mathematical Society Australian Mathematical Society lecture series 3 (DE-604)BV001902595 3 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001350790&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Giles, John R. Introduction to the analysis of metric spaces Analyse fonctionnelle Espaces linéaires normés Espaces métriques Functional analysis Metric spaces Normed linear spaces Analysis (DE-588)4001865-9 gnd Metrischer Raum (DE-588)4169745-5 gnd Normierter Raum (DE-588)4127735-1 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4169745-5 (DE-588)4127735-1 |
| title | Introduction to the analysis of metric spaces |
| title_auth | Introduction to the analysis of metric spaces |
| title_exact_search | Introduction to the analysis of metric spaces |
| title_full | Introduction to the analysis of metric spaces J. R. Giles |
| title_fullStr | Introduction to the analysis of metric spaces J. R. Giles |
| title_full_unstemmed | Introduction to the analysis of metric spaces J. R. Giles |
| title_short | Introduction to the analysis of metric spaces |
| title_sort | introduction to the analysis of metric spaces |
| topic | Analyse fonctionnelle Espaces linéaires normés Espaces métriques Functional analysis Metric spaces Normed linear spaces Analysis (DE-588)4001865-9 gnd Metrischer Raum (DE-588)4169745-5 gnd Normierter Raum (DE-588)4127735-1 gnd |
| topic_facet | Analyse fonctionnelle Espaces linéaires normés Espaces métriques Functional analysis Metric spaces Normed linear spaces Analysis Metrischer Raum Normierter Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001350790&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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