Elementary functional analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Boston
De Gruyter
[2018]
|
| Series: | De Gruyter Graduate
|
| Subjects: | |
| Online Access: | http://www.degruyter.com/search?f_0=isbnissn&q_0=9783110613919&searchTitles=true https://www.degruyter.com/doc/cover/9783110613919.jpg Inhaltsverzeichnis |
| Physical Description: | XV, 314 Seiten Diagramme 24 cm x 17 cm |
| ISBN: | 9783110613919 3110613913 |
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
PREFACE* VII
1 PRELIMINARIES * 1
1.1 SET THEORETIC BASICS * 1
1.1.1 SOME TERMINOLOGY AND NOTATIONS * 1
1.1.2 CARDINALITY AND COUNTABILITY * 2
1.2 TERMINOLOGY RELATED TO FUNCTIONS * 4
1.3 UPPER AND LOWER LIMITS * 6
2 METRIC SPACES * 7
2.1 DEFINITION AND EXAMPLES * 7
2.2 HOLDER*S AND MINKOWSKI*S INEQUALITIES * 9
2.2.1 CONJUGATE INDICES * 9
2.2.2 YOUNG*S INEQUALITY * 9
2.2.3 THE CASE OF N-TUPLES * 10
2.2.4 SEQUENTIAL CASE * 12
2.3 SUBSPACES OF A METRIC SPACE * 14
2.4 FUNCTION SPACES * 14
2.5 FURTHER PROPERTIES OF METRIC * 16
2.6 CONVERGENCE AND CONTINUITY * 17
2.6.1 CONVERGENCE OF A SEQUENCE * 17
2.6.2 CONTINUITY, UNIFORM CONTINUITY, AND LIPSCHITZ CONTINUITY * 18
2.7 BALLS, SEPARATION, AND BOUNDEDNESS * 20
2.8 INTERIOR POINTS, OPEN SETS
-----
23
2.9 LIMIT POINTS, CLOSED SETS * 24
2.10 DENSE SETS AND SEPARABLE SPACES * 27
2.11 EXTERIOR AND BOUNDARY * 28
2.12 EQUIVALENT METRICS, HOMEOMORPHISMS AND ISOMETRIES * 29
2.12.1 EQUIVALENT METRICS * 29
2.12.2 HOMEOMORPHISMS AND ISOMETRIES * 30
2.13 COMPLETENESS AND COMPLETION * 31
2.13.1 CAUCHY/FUNDAMENTAL SEQUENCES * 31
2.13.2 COMPLETE METRIC SPACES * 33
2.13.3 SUBSPACES OF COMPLETE METRIC SPACES * 37
2.13.4 NESTED BALLS THEOREM * 37
2.13.5 COMPLETION * 40
2.14 CATEGORY AND BAIRE CATEGORY THEOREM * 43
2.14.1 NOWHERE DENSENESS * 43
2.14.2 CATEGORY * 46
2.14.3 BAIRE CATEGORY THEOREM * 47
2.15 COMPACTNESS * 49
2.15.1 TOTAL BOUNDEDNESS * 50
2.15.2 COMPACTNESS, PRECOMPACTNESS * 54
2.15.3 HAUSDORFF CRITERION * 60
2.15.4 COMPACTNESS IN CERTAIN COMPLETE METRIC SPACES * 62
2.15.5 OTHER FORMS OF COMPACTNESS * 64
2.15.6 EQUIVALENCE OF DIFFERENT FORMS OF COMPACTNESS
-----
65
2.15.7 COMPACTNESS AND CONTINUITY * 67
2.16 SPACE (C(X,
Y),PCO)
* 70
2.17 ARZELAE-ASCOLI THEOREM * 72
2.17.1 UNIFORM BOUNDEDNESS AND EQUICONTINUITY * 72
2.17.2 ARZELAE-ASCOLI THEOREM * 73
2.17.3 APPLICATION: PEANO*S EXISTENCE THEOREM * 76
2.18 STONE-WEIERSTRASS THEOREM * 79
2.18.1 WEIERSTRASS APPROXIMATION THEOREM * 79
2.18.2 ALGEBRAS * 79
2.18.3 STONE-WEIERSTRASS THEOREM * 82
2.18.4 APPLICATIONS * 87
2.19 PROBLEMS * 87
3 NORMED VECTOR AND BANACH SPACES * 97
3.1 VECTOR SPACES * 97
3.1.1 DEFINITION, EXAMPLES, PROPERTIES * 97
3.1.2 HOMOMORPHISMS AND ISOMORPHISMS * 100
3.1.3 SUBSPACES * 101
3.1.4 SPANS AND LINEAR COMBINATIONS * 103
3.1.5 LINEAR INDEPENDENCE, HAMEL BASES, DIMENSION * 103
3.1.6 NEW SPACES FROM OLD * 108
3.1.7 DISJOINT AND COMPLEMENTARY SUBSPACES, DIRECT SUM DECOMPOSITIONS,
DEFICIENCY AND CODIMENSION * 111
3.2 NORMED VECTOR AND BANACH SPACES * 114
3.2.1 DEFINITIONS AND EXAMPLES * 114
3.2.2 SERIES AND COMPLETENESS CHARACTERIZATION * 118
3.2.3 COMPARING NORMS, EQUIVALENT NORMS * 119
3.2.4 ISOMETRIC ISOMORPHISMS * 120
3.2.5 COMPLETION * 121
3.2.6 TOPOLOGICAL AND SCHAUDER BASES * 122
3.3 FINITE-DIMENSIONAL SPACES AND RELATED TOPICS * 125
3.3.1 NORM EQUIVALENCE AND COMPLETENESS
-----
125
3.3.2 FINITE-DIMENSIONAL SUBSPACES AND BASES OF BANACH SPACES * 127
3.4 RIESZ S LEMMA AND IMPLICATIONS * 130
3.5 CONVEXITY, STRICTLY CONVEX NORMED VECTOR SPACES * 132
3.5.1 CONVEXITY * 132
3.5.2 STRICTLY CONVEX NORMED VECTOR SPACES * 133
3.6 PROBLEMS * 137
4 INNER PRODUCT AND HILBERT SPACES * 141
4.1 DEFINITIONS AND EXAMPLES * 141
4.2 INNER PRODUCT NORM, CAUCHY-SCHWARZ INEQUALITY * 143
4.3 HILBERT SPACES * 145
4.4 CERTAIN GEOMETRIC PROPERTIES * 147
4.4.1 POLARIZATION IDENTITIES * 147
4.4.2 PARALLELOGRAM LAW * 147
4.4.3 ORTHOGONALITY * 151
4.5 NEAREST POINT PROPERTY * 152
4.6 PROJECTION THEOREM * 155
4.6.1 ORTHOGONAL COMPLEMENTS * 155
4.6.2 PROJECTION THEOREM * 157
4.7 COMPLETION * 159
4.8 GRAM DETERMINANT * 160
4.9 ORTHOGONAL AND ORTHONORMAL SETS * 165
4.10 GRAM-SCHMIDT PROCESS * 170
4.11 GENERALIZED FOURIER SERIES * 171
4.11.1 FINITE ORTHONORMAL SET* 172
4.11.2 ARBITRARY ORTHONORMAL SET * 174
4.11.3 ORTHONORMAL SEQUENCE * 178
4.12 ORTHONORMAL BASES AND ORTHOGONAL DIMENSION * 179
4.13 PROBLEMS * 184
5 LINEAR OPERATORS AND FUNCTIONALS * 187
5.1 LINEAR OPERATORS AND FUNCTIONALS * 187
5.1.1 DEFINITIONS AND EXAMPLES * 187
5.1.2 KERNEL, RANGE, AND GRAPH * 189
5.1.3 RANK-NULLITY AND EXTENSION THEOREMS * 189
5.2 BOUNDED LINEAR OPERATORS AND FUNCTIONALS * 192
5.2.1 DEFINITIONS, PROPERTIES, AND EXAMPLES * 192
5.2.2 SPACE OF BOUNDED LINEAR OPERATORS, DUAL SPACE * 197
5.3 CLOSED LINEAR OPERATORS * 201
5.4 PROBLEMS * 204
6 THREE FUNDAMENTAL PRINCIPLES OF LINEAR FUNCTIONAL ANALYSIS * 207
6.1 HAHN-BANACH THEOREM * 207
6.1.1 HAHN-BANACH THEOREM FOR REAL VECTOR SPACES * 207
6.1.2 HAHN-BANACH THEOREM FOR NORMED VECTOR SPACES * 210
6.2 IMPLICATIONS OF THE HAHN-BANACH THEOREM * 212
6.2.1 SEPARATION AND NORM REALIZATION * 212
6.2.2 CHARACTERIZATION OF FUNDAMENTALITY * 215
6.2.3 SUFFICIENCY FOR SEPARABILITY * 216
6.2.4 ISOMETRIC EMBEDDING THEOREMS
-----
217
6.2.5 SECOND DUAL SPACE AND CANONICAL ISOMORPHISM * 218
6.2.6 CLOSED COMPLEMENTED SUBSPACES * 219
6.3 WEAK AND WEAK* CONVERGENCE * 221
6.4 UNIFORM BOUNDEDNESS PRINCIPLE, THE BANACH-STELNHAUS
THEOREM
-----
225
6.4.1 UNIFORM BOUNDEDNESS PRINCIPLE * 225
6.4.2 BANACH-STEINHAUS THEOREM * 228
6.5 APPLICATIONS OF THE UNIFORM BOUNDEDNESS PRINCIPLE * 231
6.5.1 WEAK BOUNDEDNESS * 232
6.5.2 MATRIX METHODS OF CONVERGENCE AND SUMMABILITY * 233
6.6 OPEN MAPPING, INVERSE MAPPING, AND CLOSED GRAPH THEOREMS * 240
6.6.1 OPEN MAPPING THEOREM * 240
6.6.2 INVERSE MAPPING THEOREM AND APPLICATIONS * 245
6.6.3 CLOSED GRAPH THEOREM AND APPLICATION * 249
6.6.4 EQUIVALENCE OF
OMT, IMT,
AND
C6T
* 255
6.7 PROBLEMS
-----
256
7 DUALITY AND REFLEXIV!TY * 261
7.1 SELF-DUALITY OF HILBERT SPACES * 261
7.1.1 RIESZ REPRESENTATION THEOREM * 261
7.1.2 LINEAR BOUNDED FUNCTIONALS ON CERTAIN HILBERT SPACES * 264
7.1.3 WEAK CONVERGENCE IN HILBERT SPACES * 264
7.2 DUALITY OF FINITE-DIMENSIONAL SPACES * 265
7.2.1 REPRESENTATION THEOREM * 265
7.2.2 WEAK CONVERGENCE IN FINITE-DIMENSIONAL SPACES * 268
7.3 DUALITY OF SEQUENCE SPACES * 268
7.3.1 REPRESENTATION THEOREM FOR /* * 268
7.3.2 WEAK CONVERGENCE IN
LP
(1 P OO)
* 272
7.3.3 DUALITY AND WEAK CONVERGENCE FOR (C0, || - I
CQ
) * 276
7.4 DUALITY AND WEAK CONVERGENCE FOR (C[O,
B],
|| * H^) * 277
7.4.1 RIESZ REPRESENTATION THEOREM FOR
C [A, B]
* 277
7.4.2 WEAK CONVERGENCE IN
(C[A, B],
|| * * 278
7.5 REFLEXIVITY * 278
7.5.1 DEFINITION AND EXAMPLES
-----
278
7.5.2 COMPLETENESS OF A REFLEXIVE SPACE * 283
7.5.3 REFLEXIVITY OF A CLOSED SUBSPACE * 283
7.5.4 ISOMETRIC ISOMORPHISM AND REFLEXIVITY * 285
7.5.5 CHARACTERIZATION OF REFLEXIVITY * 286
7.5.6 WEAK CONVERGENCE AND WEAK COMPLETENESS * 287
7.5.7 BOUNDED SEQUENCE PROPERTY * 288
7.6 PROBLEMS * 290
A THE AXIOM OF CHOICE AND EQUIVALENTS * 293
A .L THE AXIOM OF CHOICE * 293
A. 1.1 THE AXIOM OF CHOICE * 293
A .L.2 CONTROVERSY * 293
A.1.3 TIMELINE * 294
A.2 ORDERED SETS * 294
A.3 EQUIVALENTS * 298
BIBLIOGRAPHY * 303
INDEX* 305
|
| any_adam_object | 1 |
| author | Markin, Marat V. |
| author_GND | (DE-588)1169968635 |
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| bvnumber | BV045231755 |
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| ctrlnum | (OCoLC)1059559738 (DE-599)DNB1162289376 |
| discipline | Mathematik |
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| spelling | Markin, Marat V. Verfasser (DE-588)1169968635 aut Elementary functional analysis Marat V. Markin Berlin ; Boston De Gruyter [2018] © 2018 XV, 314 Seiten Diagramme 24 cm x 17 cm txt rdacontent n rdamedia nc rdacarrier De Gruyter Graduate Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Funktionalanalysis (DE-588)4018916-8 s DE-604 Walter de Gruyter GmbH & Co. KG (DE-588)10095502-2 pbl Erscheint auch als Online-Ausgabe, EPUB 978-3-11-061409-1 Erscheint auch als Online-Ausgabe, PDF 978-3-11-061403-9 X:MVB http://www.degruyter.com/search?f_0=isbnissn&q_0=9783110613919&searchTitles=true X:MVB https://www.degruyter.com/doc/cover/9783110613919.jpg DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030620125&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Markin, Marat V. Elementary functional analysis Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4018916-8 (DE-588)4123623-3 |
| title | Elementary functional analysis |
| title_auth | Elementary functional analysis |
| title_exact_search | Elementary functional analysis |
| title_full | Elementary functional analysis Marat V. Markin |
| title_fullStr | Elementary functional analysis Marat V. Markin |
| title_full_unstemmed | Elementary functional analysis Marat V. Markin |
| title_short | Elementary functional analysis |
| title_sort | elementary functional analysis |
| topic | Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Funktionalanalysis Lehrbuch |
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