Real and functional analysis:
The second edition was published as Real Analysis, Addison-Wesley, 1983. The third edition has been reorganized. After a brief introduction to point set topology, some basic theorems on continuous functions, and the introduction of Banach and Hilbert spaces, integration theory is covered so it can f...
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
1993
|
| Edition: | 3. ed. |
| Series: | Graduate texts in mathematics
142 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | The second edition was published as Real Analysis, Addison-Wesley, 1983. The third edition has been reorganized. After a brief introduction to point set topology, some basic theorems on continuous functions, and the introduction of Banach and Hilbert spaces, integration theory is covered so it can fit in a one-semester course. |
| Item Description: | Frühere Aufl. u.d.T.: Lang, Serge: Real analysis |
| Physical Description: | XIV, 580 S. graph. Darst. |
| ISBN: | 0387940014 3540940014 |
Staff View
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Record in the Search Index
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| adam_text | SERGE LANG REAL AND FUNCTIONAL ANALYSIS THIRD EDITION WITH 37
ILLUSTRATIONS SPRINGER CONTENTS PART ONE GENERAL TOPOLOGY . 1
CHAPTER I SETS 3 §1. SOME BASIC TERMINOLOGY 3 §2. DENUMERABLE SETS 7 §3.
ZORN S LEMMA 10 CHAPTER II TOPOLOGICAL SPACES 17 §1. OPEN AND CLOSED
SETS 17 §2. CONNECTED SETS 27 §3. COMPACT SPACES 31 §4. SEPARATION BY
CONTINUOUS FUNCTIONS 40 §5. EXERCISES 43 CHAPTER III CONTINUOUS
FUNCTIONS ON COMPACT SETS 51 §1. THE STONE-WEIERSTRASS THEOREM 51 §2.
IDEALS OF CONTINUOUS FUNCTIONS 55 §3. ASCOLI S THEOREM 57 §4. EXERCISES
59 X CONTENTS PART TWO BANACH AND HILBERT SPACES 63 CHAPTER IV BANACH
SPACES 65 §1. DEFINITIONS, THE DUAL SPACE, AND THE HAHN-BANACH THEOREM
65 §2. BANACH ALGEBRAS 72 §3. THE LINEAR EXTENSION THEOREM 75 §4.
COMPLETION OF A NORMED VECTOR SPACE 76 §5. SPACES WITH OPERATORS 81
APPENDIX: CONVEX SETS 83 1. THE KREIN-MILMAN THEOREM 83 2. MAZUR S
THEOREM 88 §6. EXERCISES 91 CHAPTER V HILBERT SPACE 95 §1. HERMITIAN
FORMS 95 §2. FUNCTIONALS AND OPERATORS 104 §3. EXERCISES 107 PART
THREE INTEGRATION 109 CHAPTER VI THE GENERAL INTEGRAL ILL §1. MEASURED
SPACES, MEASURABLE MAPS, AND POSITIVE MEASURES 112 §2. THE INTEGRAL OF
STEP MAPS 126 §3. THE ^-COMPLETION 128 §4. PROPERTIES OF THE INTEGRAL:
FIRST PART 134 §5. PROPERTIES OF THE INTEGRAL: SECOND PART 137 §6.
APPROXIMATIONS 147 §7. EXTENSION OF POSITIVE MEASURES FROM ALGEBRAS TO
FF-ALGEBRAS 153 §8. PRODUCT MEASURES AND INTEGRATION ON A PRODUCT SPACE
158 §9. THE LEBESGUE INTEGRAL IN R P 166 §10. EXERCISES 172 CHAPTER
VII DUALITY AND REPRESENTATION THEOREMS 181 §1. THE HILBERT SPACE L 2
(N) 181 §2. DUALITY BETWEEN L L (IX) AND L(/I) 185 §3. COMPLEX AND
VECTORIAL MEASURES 195 §4. COMPLEX OR VECTORIAL MEASURES AND DUALITY 204
§5. THE L SPACES, 1 P OO 209 §6. THE LAW OF LARGE NUMBERS 213 §7.
EXERCISES 217 CONTENTS XI CHAPTER VIII SOME APPLICATIONS OF INTEGRATION
223 §1. CONVOLUTION 223 §2. CONTINUITY AND DIFFERENTIATION UNDER THE
INTEGRAL SIGN 225 §3. DIRAC SEQUENCES 227 §4. THE SCHWARTZ SPACE AND
FOURIER TRANSFORM 236 §5. THE FOURIER INVERSION FORMULA 241 §6. THE
POISSON SUMMATION FORMULA 243 §7. AN EXAMPLE OF FOURIER TRANSFORM NOT IN
THE SCHWARTZ SPACE 244 §8. EXERCISES 247 CHAPTER IX INTEGRATION AND
MEASURES ON LOCALLY COMPACT SPACES 251 §1. POSITIVE AND BOUNDED
FUNCTIONALS ON C C (X) 252 §2. POSITIVE FUNCTIONALS AS INTEGRALS 255 §3.
REGULAR POSITIVE MEASURES 265 §4. BOUNDED FUNCTIONALS AS INTEGRALS 267
§5. LOCALIZATION OF A MEASURE AND OF THE INTEGRAL 269 §6. PRODUCT
MEASURES ON LOCALLY COMPACT SPACES 272 §7. EXERCISES 274 CHAPTER X
RIEMANN-STIELTJES INTEGRAL AND MEASURE 278 §1. FUNCTIONS OF BOUNDED
VARIATION AND THE STIELTJES INTEGRAL 278 §2. APPLICATIONS TO FOURIER
ANALYSIS 287 §3. EXERCISES . 294 CHAPTER XI DISTRIBUTIONS 295 §1.
DEFINITION AND EXAMPLES 295 §2. SUPPORT AND LOCALIZATION 299 §3.
DERIVATION OF DISTRIBUTIONS 303 §4. DISTRIBUTIONS WITH DISCRETE SUPPORT
304 CHAPTER XII INTEGRATION ON LOCALLY COMPACT GROUPS 308 §1.
TOPOLOGICAL GROUPS 308 §2. THE HAAR INTEGRAL, UNIQUENESS 313 §3.
EXISTENCE OF THE HAAR INTEGRAL 319 §4. MEASURES ON FACTOR GROUPS AND
HOMOGENEOUS SPACES 322 §5. EXERCISES 326 PART FOUR CALCULUS 329 XLL
CONTENTS CHAPTER XIII DIFFERENTIAL CALCULUS 331 §1. INTEGRATION IN ONE
VARIABLE 331 §2. THE DERIVATIVE AS A LINEAR MAP 333 §3. PROPERTIES OF
THE DERIVATIVE 335 §4. MEAN VALUE THEOREM 340 §5. THE SECOND DERIVATIVE
343 §6. HIGHER DERIVATIVES AND TAYLOR S FORMULA 346 §7. PARTIAL
DERIVATIVES 351 §8. DIFFERENTIATING UNDER THE INTEGRAL SIGN 355 §9.
DIFFERENTIATION OF SEQUENCES 356 §10. EXERCISES 357 CHAPTER XIV INVERSE
MAPPINGS AND DIFFERENTIAL EQUATIONS 360 §1. THE INVERSE MAPPING THEOREM
360 §2. THE IMPLICIT MAPPING THEOREM 364 §3. EXISTENCE THEOREM FOR
DIFFERENTIAL EQUATIONS 365 §4. LOCAL DEPENDENCE ON INITIAL CONDITIONS
371 §5. GLOBAL SMOOTHNESS OF THE FLOW 376 §6. EXERCISES 379 PART FIVE
FUNCTIONAL ANALYSIS 385 CHAPTER XV THE OPEN MAPPING THEOREM, FACTOR
SPACES, AND DUALITY 387 §1. THE OPEN MAPPING THEOREM 387 §2.
ORTHOGONALITY 391 §3. APPLICATIONS OF THE OPEN MAPPING THEOREM 395
CHAPTER XVI THE SPECTRUM 400 §1. THE GELFAND-MAZUR THEOREM . 400 §2.
THE GELFAND TRANSFORM .^ 407 §3. C*-ALGEBRAS 409 §4. EXERCISES 412
CHAPTER XVII COMPACT AND FREDHOLM OPERATORS 415 §1. COMPACT OPERATORS
415 §2. FREDHOLM OPERATORS A HD THE INDEX 417 §3. SPECTRAL THEOREM FOR
COMPACT OPERATORS 426 §4. APPLICATION TO INTEGRAL EQUATIONS 432 §5.
EXERCISES 433 CONTENTS X1U CHAPTER XVIII SPECTRAL THEOREM FOR BOUNDED
HERMITIAN OPERATORS 438 §1. HERMITIAN AND UNITARY OPERATORS 438 §2.
POSITIVE HERMITIAN OPERATORS 439 §3. THE SPECTRAL THEOREM FOR COMPACT
HERMITIAN OPERATORS 442 §4. THE SPECTRAL THEOREM FOR HERMITIAN OPERATORS
444 §5. ORTHOGONAL PROJECTIONS 449 §6. SCHUR S LEMMA 452 §7. POLAR
DECOMPOSITION OF ENDOMORPHISMS 453 §8. THE MORSE-PALAIS LEMMA 455 §9.
EXERCISES 458 CHAPTER XIX FURTHER SPECTRAL THEOREMS 464 §1. PROJECTION
FUNCTIONS OF OPERATORS 464 §2. SELF-ADJOINT OPERATORS 469 §3. EXAMPLE:
THE LAPLACE OPERATOR IN THE PLANE 476 CHAPTER XX SPECTRAL MEASURES 480
§1. DEFINITION OF THE SPECTRAL MEASURE 480 §2. UNIQUENESS OF THE
SPECTRAL MEASURE: THE TITCHMARSH-KODAIRA FORMULA 485 §3. UNBOUNDED
FUNCTIONS OF OPERATORS 488 §4. SPECTRAL FAMILIES OF PROJECTIONS 490 §5.
THE SPECTRAL INTEGRAL AS STIELTJES INTEGRAL 491 §6. EXERCISES 492 PART
SIX GLOBAL ANALYSIS 495 CHAPTER XXI LOCAL INTEGRATION OF DIFFERENTIAL
FORMS 497 §1. SETS OF MEASURE 0 497 §2. CHANGE OF VARIABLES FORMULA ,
498 §3. DIFFERENTIAL FORMS 507 §4. INVERSE IMAGE OF A FORM *? 512 §5.
APPENDIX 516 CHAPTER XXII MANIFOLDS 523 §1. ATLASES, CHARTS, MORPHISMS
523 §2. SUBMANIFOLDS 527 §3. TANGENT SPACES ... F . 533 §4. PARTITIONS
OF UNITY 536 §5. MANIFOLDS WITH BOUNDARY 539 §6. VECTOR FIELDS AND
GLOBAL DIFFERENTIAL EQUATIONS , 543 XIV CONTENTS CHAPTER XXIII
INTEGRATION AND MEASURES ON MANIFOLDS 547 §1. DIFFERENTIAL FORMS ON
MANIFOLDS 547 §2. ORIENTATION 551 §3. THE MEASURE ASSOCIATED WITH A
DIFFERENTIAL FORM 553 §4. STOKES THEOREM FOR A RECTANGULAR SIMPLEX 555
§5. STOKES THEOREM ON A MANIFOLD 558 §6. STOKES THEOREM WITH
SINGULARITIES 561 BIBLIOGRAPHY 569 TABLE OF NOTATION 572 INDEX 575
|
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| indexdate | 2024-07-09T17:08:31Z |
| institution | BVB |
| isbn | 0387940014 3540940014 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005081359 |
| oclc_num | 243743114 |
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| physical | XIV, 580 S. graph. Darst. |
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| publisher | Springer |
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| series2 | Graduate texts in mathematics |
| spelling | Lang, Serge 1927-2005 Verfasser (DE-588)119305119 aut Real and functional analysis Serge Lang 3. ed. New York, NY [u.a.] Springer 1993 XIV, 580 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 142 Frühere Aufl. u.d.T.: Lang, Serge: Real analysis The second edition was published as Real Analysis, Addison-Wesley, 1983. The third edition has been reorganized. After a brief introduction to point set topology, some basic theorems on continuous functions, and the introduction of Banach and Hilbert spaces, integration theory is covered so it can fit in a one-semester course. Mathematical analysis Analysis (DE-588)4001865-9 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 s DE-604 Analysis (DE-588)4001865-9 s Erscheint auch als Online-Ausgabe 978-1-4612-0897-6 (DE-604)BV042419666 Graduate texts in mathematics 142 (DE-604)BV000000067 142 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005081359&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lang, Serge 1927-2005 Real and functional analysis Graduate texts in mathematics Mathematical analysis Analysis (DE-588)4001865-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4018916-8 |
| title | Real and functional analysis |
| title_auth | Real and functional analysis |
| title_exact_search | Real and functional analysis |
| title_full | Real and functional analysis Serge Lang |
| title_fullStr | Real and functional analysis Serge Lang |
| title_full_unstemmed | Real and functional analysis Serge Lang |
| title_short | Real and functional analysis |
| title_sort | real and functional analysis |
| topic | Mathematical analysis Analysis (DE-588)4001865-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Mathematical analysis Analysis Funktionalanalysis |
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