Real analysis:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Prentice Hall
2010
|
| Edition: | 4. ed., internat. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 505 S. |
| ISBN: | 9780135113554 0135113555 |
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MARC
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Record in the Search Index
| _version_ | 1804141145944489984 |
|---|---|
| adam_text | Contents
Preface
ix
I Lebesgue Integration for Functions of a Single Real Variable
1
Preliminaries on Sets, Mappings, and Relations
3
Unions and Intersections of Sets
............................. 3
Equivalence Relations, the Axiom of Choice, and Zorn s Lemma
.......... 5
1
The Real Numbers: Sets, Sequences, and Functions
7
1.1
The Field.
Positivity,
and Completeness Axioms
................. 7
1.2
The Natural and Rational Numbers
........................ 11
1.3
Countable and Uncountable Sets
......................... 13
1.4
Open Sets, Closed Sets, and
Borei
Sets of Real Numbers
............ 16
1.5
Sequences of Real Numbers
............................ 20
1.6
Continuous Real-Valued Functions of a Real Variable
............. 25
2
Lebesgue Measure
29
2.1
Introduction
..................................... 29
2.2
Lebesgue Outer Measure
.............................. 31
2.3
The
σ-
Algebra of Lebesgue Measurable Sets
.................. 34
2.4
Outer and Inner Approximation of Lebesgue Measurable Sets
........ 40
2.5
Countable Additivity. Continuity, and the
Borei-Cantelli
Lemma
....... 43
2.6
Nonmeasurable Sets
................................. 47
2.7
The Cantor Set and the Cantor-Lebesgue Function
............... 49
3
Lebesgue Measurable Functions
54
3.1
Sums, Products, and Compositions
........................ 54
3.2
Sequential Pointwise Limits and Simple Approximation
............ 60
3.3
Littlewood s Three Principles, Egoroff
s
Theorem, and Lusin s Theorem
... 64
4
Lebesgue Integration
68
4.1
The Riemann Integral
................................ 68
4.2
The Lebesgue Integral of a Bounded Measurable Function over a Set of
Finite Measure
.................................... 71
4.3
The Lebesgue Integral of a Measurable
Nonnegative
Function
........ 79
4.4
The General Lebesgue Integral
.......................... 85
4.5
Countable Additivity and Continuity of Integration
............... 90
4.6
Uniform Integrability: The
Vitali
Convergence Theorem
............ 92
vi
Contents
5 Lebesgue Integration:
Further Topics
97
5.1 Uniform Integrability
and Tightness:
A
General
Vitali
Convergence
Theorem 97
5.2
Convergence in Measure
.............................. 99
5.3
Characterizations of
Riemann and Lebesgue Integrability........... 102
6
Differentiation and Integration
107
6.1
Continuity of Monotone Functions
........................ 108
6.2
Differentiability of Monotone Functions: Lebesgue s Theorem
........ 109
6.3
Functions of Bounded Variation: Jordan s Theorem
.............. 116
6.4
Absolutely Continuous Functions
......................... 119
6.5
Integrating Derivatives: Differentiating Indefinite Integrals
.......... 124
6.6
Convex Functions
.................................. 130
7
The IP Spaces: Completeness and Approximation
135
7.1
Normed Linear Spaces
............................... 135
7.2
The Inequalities of Young, Holder, and Minkowski
............... 139
7.3
LP Is Complete: The Riesz-Fischer Theorem
.................. 144
7.4
Approximation and Separability
.......................... 150
8
The LP Spaces: Duality and Weak Convergence
155
8.1
The Riesz Representation for the Dual of Lp A
<
ρ
<
oc ...........
155
8.2
Weak Sequential Convergence in V
....................... 162
8.3
Weak Sequential Compactness
........................... 171
8.4
The Minimization of Convex Functionals
..................... 174
II Abstract Spaces: Metric,
Topologica!, Banach,
and Hubert Spaces
181
9
Metric Spaces: General Properties
183
9.1
Examples of Metric Spaces
............................. 183
9.2
Open Sets. Closed Sets, and Convergent Sequences
............... 187
9.3
Continuous Mappings Between Metric Spaces
.................. 190
9.4
Complete Metric Spaces
.............................. 193
9.5
Compact Metric Spaces
............................... 197
9.6
Separable Metric Spaces
.............................. 204
10
Metric Spaces: Three Fundamental Theorems
206
10.1
The
Arzelà-Ascoli
Theorem
............................ 206
10.2
The Baire Category Theorem
........................... 211
10.3
The Banach Contraction Principle
......................... 215
11
Topologica]
Spaces: General Properties
222
11.1
Open Sets. Closed Sets. Bases, and
Subbases
................... 222
11.2
The Separation Properties
............................. 227
11.3
Countability and Separability
........................... 228
11.4
Continuous Mappings Between Topological Spaces
............... 230
Contents
vii
11.5
Compact
Topological Spaces............................ 233
11.6
Connected
Topological Spaces........................... 237
12
Topologica!
Spaces:
Three
Fundamental Theorems 239
12.1 Urysohn s Lemma
and the Tietze Extension
Theorem............. 239
12.2
The Tychonoff Product
Theorem......................... 244
12.3
The Stone-
Weierstrass
Theorem.......................... 247
13
Continuous
Linear Operators
Between Banach
Spaces 253
13.1 Normed Linear Spaces............................... 253
13.2 Linear Operators .................................. 256
13.3
Compactness Lost:
Infinite
Dimensional
Normed Linear Spaces........ 259
13.4
The Open Mapping and Closed Graph Theorems
................ 263
13.5
The Uniform Boundedness Principle
....................... 268
14
Duality for Normed Linear Spaces
271
14.1
Linear Functionals, Bounded Linear Functionals, and Weak Topologies
. . . 271
14.2
The Hahn-Banach Theorem
............................ 277
14.3
Reflexive Banach Spaces and Weak Sequential Convergence
......... 282
14.4
Locally Convex Topological Vector Spaces
.................... 286
14.5
The Separation of Convex Sets and Mazur s Theorem
............. 290
14.6
The Krein-Milman Theorem
............................ 295
15
Compactness Regained: The Weak Topology
298
15.1
Alaoglu s Extension of Helley s Theorem
.................... 298
15.2
Reflexivity
and Weak Compactness: Kakutani s Theorem
........... 300
15.3
Compactness and Weak Sequential Compactness: The
Eberlein-Šmulian
Theorem
....................................... 302
15.4
Metrizability of Weak Topologies
......................... 305
16
Continuous Linear Operators on Hubert Spaces
308
16.1
The Inner Product and Orthogonality
....................... 309
16.2
The Dual Space and Weak Sequential Convergence
.............. 313
16.3
Bessel s Inequality and
Orthonormal
Bases
................... 316
16.4
Adjoints
and Symmetry for Linear Operators
.................. 319
16.5
Compact Operators
................................. 324
16.6
The Hilbert-Schmidt Theorem
........................... 326
16.7
The Riesz-Schauder Theorem: Characterization of
Fredholm
Operators
. . . 329
III Measure and Integration: General Theory
335
17
General Measure Spaces: Their Properties and Construction
337
17.1
Measures and Measurable Sets
........................... 337
17.2
Signed Measures: The
Hahn
and Jordan Decompositions
........... 342
17.3
The
Carathéodory
Measure Induced by an Outer Measure
........... 346
vüi Contents
17.4
The Construction of Outer Measures
....................... 349
17.5
The
Carathéodory-Hahn
Theorem: The Extension of a Premeasure to a
Measure
....................................... 352
18
Integration Over General Measure Spaces
359
18.1
Measurable Functions
................................ 359
18.2
Integration of
Nonnegative
Measurable Functions
............... 365
18.3
Integration of General Measurable Functions
.................. 372
18.4
The Radon-Nikodym Theorem
.......................... 381
18.5
The Nikodym Metric Space: The Vitali-Hahn-Saks Theorem
......... 388
19
General Lp Spaces: Completeness, Duality, and Weak Convergence
394
19.1
The Completeness of LP{X,
μ),
1 <
ρ
<
oo
................... 394
19.2
The Riesz Representation Theorem for the Dual of LP(X,
μ),
1 <
ρ
<
oo
. . 399
19.3
The Kantorovitch Representation Theorem for the Dual of L°°(X,
μ)
.... 404
19.4
Weak Sequential Compactness in LP(X,
μ),
I
<
p<
1............. 407
19.5
Weak Sequential Compactness in
^(Χ,μ):
The Dunford-Pettis Theorem
. . 409
20
The Construction of Particular Measures
414
20.1
Product Measures: The Theorems of Fubini and Tonelli
............ 414
20.2
Lebesgue Measure on Euclidean Space R
.................... 424
20.3
Cumulative Distribution Functions and
Borei
Measures on
R
......... 437
20.4
Carathéodory
Outer Measures and Hausdorff Measures on a Metric Space
. 441
21
Measure and Topology
446
21.1
Locally Compact Topological Spaces
....................... 447
21.2
Separating Sets and Extending Functions
..................... 452
21.3
The Construction of Radon Measures
....................... 454
21.4
The Representation of Positive Linear Functionals on C^X): The Riesz-
Markov Theorem
.................................. 457
21.5
The Riesz Representation Theorem for the Dual of C{X)
........... 462
21.6
Regularity Properties of Baire Measures
..................... 470
22
Invariant Measures
477
22.1
Topological Groups: The General Linear Group
................ 477
22.2
Kakutani s Fixed Point Theorem
......................... 480
22.3
Invariant
Borei
Measures on Compact Groups:
von
Neumann s Theorem
. . 485
22.4
Measure Preserving Transformations and Ergodicity: The Bogoliubov-Krilov
Theorem
....................................... 488
Bibliography
495
Index
497
|
| any_adam_object | 1 |
| author | Royden, Halsey L. 1928-1993 Fitzpatrick, Patrick 1946- |
| author_GND | (DE-588)172349532 (DE-588)14081888X |
| author_facet | Royden, Halsey L. 1928-1993 Fitzpatrick, Patrick 1946- |
| author_role | aut aut |
| author_sort | Royden, Halsey L. 1928-1993 |
| author_variant | h l r hl hlr p f pf |
| building | Verbundindex |
| bvnumber | BV036085649 |
| callnumber-first | Q - Science |
| callnumber-label | QA331 |
| callnumber-raw | QA331.5 |
| callnumber-search | QA331.5 |
| callnumber-sort | QA 3331.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 150 SK 400 |
| ctrlnum | (OCoLC)456836719 (DE-599)BVBBV036085649 |
| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 4. ed., internat. ed. |
| format | Book |
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| open_access_boolean | |
| owner | DE-703 DE-824 DE-355 DE-BY-UBR DE-20 |
| owner_facet | DE-703 DE-824 DE-355 DE-BY-UBR DE-20 |
| physical | XII, 505 S. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Prentice Hall |
| record_format | marc |
| spelling | Royden, Halsey L. 1928-1993 Verfasser (DE-588)172349532 aut Real analysis H. L. Royden ; P. M. Fitzpatrick 4. ed., internat. ed. Boston [u.a.] Prentice Hall 2010 XII, 505 S. txt rdacontent n rdamedia nc rdacarrier Functional analysis Functions of real variables Measure theory Maßtheorie (DE-588)4074626-4 gnd rswk-swf Reelle Funktion (DE-588)4048918-8 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Reelle Variable (DE-588)4202614-3 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Analysis (DE-588)4001865-9 s DE-604 Funktion Mathematik (DE-588)4071510-3 s Reelle Variable (DE-588)4202614-3 s Funktionalanalysis (DE-588)4018916-8 s Maßtheorie (DE-588)4074626-4 s 2\p DE-604 Reelle Funktion (DE-588)4048918-8 s 3\p DE-604 Fitzpatrick, Patrick 1946- Verfasser (DE-588)14081888X aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018976612&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Royden, Halsey L. 1928-1993 Fitzpatrick, Patrick 1946- Real analysis Functional analysis Functions of real variables Measure theory Maßtheorie (DE-588)4074626-4 gnd Reelle Funktion (DE-588)4048918-8 gnd Funktionalanalysis (DE-588)4018916-8 gnd Reelle Variable (DE-588)4202614-3 gnd Analysis (DE-588)4001865-9 gnd Funktion Mathematik (DE-588)4071510-3 gnd |
| subject_GND | (DE-588)4074626-4 (DE-588)4048918-8 (DE-588)4018916-8 (DE-588)4202614-3 (DE-588)4001865-9 (DE-588)4071510-3 (DE-588)4151278-9 |
| title | Real analysis |
| title_auth | Real analysis |
| title_exact_search | Real analysis |
| title_full | Real analysis H. L. Royden ; P. M. Fitzpatrick |
| title_fullStr | Real analysis H. L. Royden ; P. M. Fitzpatrick |
| title_full_unstemmed | Real analysis H. L. Royden ; P. M. Fitzpatrick |
| title_short | Real analysis |
| title_sort | real analysis |
| topic | Functional analysis Functions of real variables Measure theory Maßtheorie (DE-588)4074626-4 gnd Reelle Funktion (DE-588)4048918-8 gnd Funktionalanalysis (DE-588)4018916-8 gnd Reelle Variable (DE-588)4202614-3 gnd Analysis (DE-588)4001865-9 gnd Funktion Mathematik (DE-588)4071510-3 gnd |
| topic_facet | Functional analysis Functions of real variables Measure theory Maßtheorie Reelle Funktion Funktionalanalysis Reelle Variable Analysis Funktion Mathematik Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018976612&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT roydenhalseyl realanalysis AT fitzpatrickpatrick realanalysis |