Real analysis

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Title:

Real analysis

Personal Author:

DiBenedetto, Emmanuele

Series:

Birkhauser advanced texts

Publication Information:

Boston : Birkh�auser, 2002

Physical Description:

xxiv, 485 p. ; 24 cm.

ISBN:

9780817642310

Subject Term:

Mathematical analysis

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010202494	QA300 D52 2002	Open Access Book	Book	Searching... Unknown
Searching... PSZ KL	30000010222044	QA300 D52 2002	Open Access Book	Book	Searching... Unknown

This graduate text in real analysis is a solid building block for research in analysis, PDEs, the calculus of variations, probability, and approximation theory. It covers all the core topics, such as a basic introduction to functional analysis, and it discusses other topics often not addressed including Radon measures, the Besicovitch covering Theorem, the Rademacher theorem, and a constructive presentation of the Stone-Weierstrass Theoroem.

Reviews 1

Choice Review

The book is a valuable, comprehensive reference source on real analysis. The first eight chapters cover core material that is part of most courses taught on the subject, followed by a collection of special topics that stay within the framework of real analysis. In addition to the content, what makes the book especially useful as a reference source is its organization. The table of contents is 16 pages long, as there is more contained information than can be found in most books on the subject. The subject index is also longer than usual. It should be noted that the book is not ideal for self-study or classroom use. There is very little text connecting one result to another or other text that puts results into context--undergraduates will find the style terse. There are not enough exercises, and their solutions are never included (however, there are occasional hints). The exercises are often placed far away from the related material in units called "Problems and Complements," but there are numerous sections without any exercises. Summing Up: Recommended. Graduate students and faculty. This work should be used solely as a reference. --Miklos Bona, University of Florida

Preface	p. xv
Acknowledgments	p. xxiii
Preliminaries	p. 1
1 Countable sets	p. 1
2 The Cantor set	p. 2
3 Cardinality	p. 4
3.1 Some examples	p. 5
4 Cardinality of some infinite Cartesian products	p. 6
5 Orderings, the maximal principle, and the axiom of choice	p. 8
6 Well-ordering	p. 9
6.1 The first uncountable	p. 11
Problems and Complements	p. 11
I Topologies and Metric Spaces	p. 17
1 Topological spaces	p. 17
1.1 Hausdorff and normal spaces	p. 19
2 Urysohn's lemma	p. 19
3 The Tietze extension theorem	p. 21
4 Bases, axioms of countability, and product topologies	p. 22
4.1 Product topologies	p. 24
5 Compact topological spaces	p. 25
5.1 Sequentially compact topological spaces	p. 26
6 Compact subsets of R[superscript N]	p. 27
7 Continuous functions on countably compact spaces	p. 29
8 Products of compact spaces	p. 30
9 Vector spaces	p. 31
9.1 Convex sets	p. 33
9.2 Linear maps and isomorphisms	p. 33
10 Topological vector spaces	p. 34
10.1 Boundedness and continuity	p. 35
11 Linear functionals	p. 36
12 Finite-dimensional topological vector spaces	p. 36
12.1 Locally compact spaces	p. 37
13 Metric spaces	p. 38
13.1 Separation and axioms of countability	p. 39
13.2 Equivalent metrics	p. 40
13.3 Pseudometrics	p. 40
14 Metric vector spaces	p. 41
14.1 Maps between metric spaces	p. 42
15 Spaces of continuous functions	p. 43
15.1 Spaces of continuously differentiable functions	p. 44
16 On the structure of a complete metric space	p. 44
17 Compact and totally bounded metric spaces	p. 46
17.1 Precompact subsets of X	p. 48
Problems and Complements	p. 49
II Measuring Sets	p. 65
1 Partitioning open subsets of R[superscript N]	p. 65
2 Limits of sets, characteristic functions, and [sigma]-algebras	p. 67
3 Measures	p. 68
3.1 Finite, [sigma]-finite, and complete measures	p. 71
3.2 Some examples	p. 71
4 Outer measures and sequential coverings	p. 72
4.1 The Lebesgue outer measure in R[superscript N]	p. 73
4.2 The Lebesgue-Stieltjes outer measure	p. 73
5 The Hausdorff outer measure in R[superscript N]	p. 74
6 Constructing measures from outer measures	p. 76
7 The Lebesgue-Stieltjes measure on R	p. 79
7.1 Borel measures	p. 80
8 The Hausdorff measure on R[superscript N]	p. 80
9 Extending measures from semialgebras to [sigma]-algebras	p. 82
9.1 On the Lebesgue-Stieltjes and Hausdorff measures	p. 84
10 Necessary and sufficient conditions for measurability	p. 84
11 More on extensions from semialgebras to [sigma]-algebras	p. 86
12 The Lebesgue measure of sets in R[superscript N]	p. 88
12.1 A necessary and sufficient condition of measurability	p. 88
13 A nonmeasurable set	p. 90
14 Borel sets, measurable sets, and incomplete measures	p. 91
14.1 A continuous increasing function f: [0, 1] [not right arrow] [0, 1]	p. 91
14.2 On the preimage of a measurable set	p. 93
14.3 Proof of Propositions 14.1 and 14.2	p. 94
15 More on Borel measures	p. 94
15.1 Some extensions to general Borel measures	p. 97
15.2 Regular Borel measures and Radon measures	p. 97
16 Regular outer measures and Radon measures	p. 98
16.1 More on Radon measures	p. 99
17 Vitali coverings	p. 99
18 The Besicovitch covering theorem	p. 103
19 Proof of Proposition 18.2	p. 105
20 The Besicovitch measure-theoretical covering theorem	p. 107
Problems and Complements	p. 110
III The Lebesgue Integral	p. 123
1 Measurable functions	p. 123
2 The Egorov theorem	p. 126
2.1 The Egorov theorem in R[superscript N]	p. 128
2.2 More on Egorov's theorem	p. 128
3 Approximating measurable functions by simple functions	p. 128
4 Convergence in measure	p. 130
5 Quasi-continuous functions and Lusin's theorem	p. 133
6 Integral of simple functions	p. 135
7 The Lebesgue integral of nonnegative functions	p. 136
8 Fatou's lemma and the monotone convergence theorem	p. 137
9 Basic properties of the Lebesgue integral	p. 139
10 Convergence theorems	p. 141
11 Absolute continuity of the integral	p. 142
12 Product of measures	p. 142
13 On the structure of (A x B)	p. 144
14 The Fubini-Tonelli theorem	p. 147
14.1 The Tonelli version of the Fubini theorem	p. 148
15 Some applications of the Fubini-Tonelli theorem	p. 148
15.1 Integrals in terms of distribution functions	p. 148
15.2 Convolution integrals	p. 149
15.3 The Marcinkiewicz integral	p. 150
16 Signed measures and the Hahn decomposition	p. 151
17 The Radon-Nikodym theorem	p. 154
18 Decomposing measures	p. 157
18.1 The Jordan decomposition	p. 157
18.2 The Lebesgue decomposition	p. 159
18.3 A general version of the Radon-Nikodym theorem	p. 160
Problems and Complements	p. 160
IV Topics on Measurable Functions of Real Variables	p. 171
1 Functions of bounded variations	p. 171
2 Dini derivatives	p. 173
3 Differentiating functions of bounded variation	p. 176
4 Differentiating series of monotone functions	p. 177
5 Absolutely continuous functions	p. 179
6 Density of a measurable set	p. 181
7 Derivatives of integrals	p. 182
8 Differentiating Radon measures	p. 184
9 Existence and measurability of D[subscript mu]v	p. 186
9.1 Proof of Proposition 9.2	p. 188
10 Representing D[subscript mu]v	p. 189
10.1 Representing D[subscript mu]v for v	p. 189
10.2 Representing D[subscript mu]v for v [perpendicular, bottom] [mu]	p. 191
11 The Lebesgue differentiation theorem	p. 191
11.1 Points of density	p. 192
11.2 Lebesgue points of an integrable function	p. 192
12 Regular families	p. 193
13 Convex functions	p. 194
14 Jensen's inequality	p. 196
15 Extending continuous functions	p. 197
16 The Weierstrass approximation theorem	p. 199
17 The Stone-Weierstrass theorem	p. 200
18 Proof of the Stone-Weierstrass theorem	p. 201
18.1 Proof of Stone's theorem	p. 202
19 The Ascoli-Arzela theorem	p. 203
19.1 Precompact subsets of C (E)	p. 204
Problems and Complements	p. 205
V The L[superscript p] (E) Spaces	p. 221
1 Functions in L[superscript p] (E) and their norms	p. 221
1.1 The spaces L[superscript p] for 0	p. 222
1.2 The spaces L[superscript q] for q	p. 222
2 The Holder and Minkowski inequalities	p. 223
3 The reverse Holder and Minkowski inequalities	p. 224
4 More on the spaces L[superscript p] and their norms	p. 225
4.1 Characterizing the norm [double vertical line]f[double vertical line subscript p] for 1 [less than or equal] p	p. 225
4.2 The norm [double vertical line middle dot double vertical line] for E of finite measure	p. 226
4.3 The continuous version of the Minkowski inequality	p. 227
5 L[superscript p] (E) for 1 [less than or equal] p [less than or equal] [infinity] as normed spaces of equivalence classes	p. 227
5.1 L[superscript p] (E) for 1 [less than or equal] p [less than or equal] [infinity] as a metric topological vector space	p. 228
6 A metric topology for L[superscript p] (E) when 0	p. 229
6.1 Open convex subsets of L[superscript p] (E) when 0	p. 229
7 Convergence in L[superscript p] (E) and completeness	p. 230
8 Separating L[superscript p] (E) by simple functions	p. 232
9 Weak convergence in L[superscript p] (E)	p. 234
9.1 A counterexample	p. 234
10 Weak lower semicontinuity of the norm in L[superscript p] (E)	p. 235
11 Weak convergence and norm convergence	p. 236
11.1 Proof of Proposition 11.1 for p [greater than or equal] 2	p. 237
11.2 Proof of Proposition 11.1 for 1	p. 237
12 Linear functionals in L[superscript p] (E)	p. 238
13 The Riesz representation theorem	p. 239
13.1 Proof of Theorem 13.1: The case where {{X, A, [mu]{{ is finite	p. 240
13.2 Proof of Theorem 13.1: The case where {{X, A, [mu]{{ is [sigma]-finite	p. 241
13.3 Proof of Theorem 13.1: The case where 1	p. 242
14 The Hanner and Clarkson inequalities	p. 243
14.1 Proof of Hanner's inequalities	p. 244
14.2 Proof of Clarkson's inequalities	p. 245
15 Uniform convexity of L[superscript p] (E) for 1	p. 246
16 The Riesz representation theorem by uniform convexity	p. 247
16.1 Proof of Theorem 13.1: The case where 1	p. 247
16.2 The case where p = 1 and E is of finite measure	p. 248
16.3 The case where p = 1 and {{X, A, [mu]{{ is [sigma]-finite	p. 249
17 Bounded linear functional in L[superscript p] (E) for 0	p. 250
17.1 An alternate proof of Proposition 17.1	p. 250
18 If E [subset or is implied by] R[superscript N] and p [set membership] [1, [infinity]), then L[superscript p] (E) is separable	p. 251
18.1 L[superscript infinity] (E) is not separable	p. 254
19 Selecting weakly convergent subsequences	p. 254
20 Continuity of the translation in L[superscript p] (E) for 1 [less than or equal] p	p. 255
21 Approximating functions in L[superscript p] (E) with functions in C[superscript infinity] (E)	p. 257
22 Characterizing precompact sets in L[superscript p] (E)	p. 260
Problems and Complements	p. 262
VI Banach Spaces	p. 275
1 Normed spaces	p. 275
1.1 Seminorms and quotients	p. 276
2 Finite- and infinite-dimensional normed spaces	p. 277
2.1 A counterexample	p. 277
2.2 The Riesz lemma	p. 278
2.3 Finite-dimensional spaces	p. 279
3 Linear maps and functionals	p. 280
4 Examples of maps and functionals	p. 282
4.1 Functionals	p. 283
4.2 Linear functionals on C (E)	p. 283
5 Kernels of maps and functionals	p. 284
6 Equibounded families of linear maps	p. 285
6.1 Another proof of Proposition 6.1	p. 286
7 Contraction mappings	p. 286
7.1 Applications to some Fredholm integral equations	p. 287
8 The open mapping theorem	p. 288
8.1 Some applications	p. 289
8.2 The closed graph theorem	p. 289
9 The Hahn-Banach theorem	p. 290
10 Some consequences of the Hahn-Banach theorem	p. 292
10.1 Tangent planes	p. 295
11 Separating convex subsets of X	p. 295
12 Weak topologies	p. 297
12.1 Weakly and strongly closed convex sets	p. 299
13 Reflexive Banach spaces	p. 300
14 Weak compactness	p. 301
14.1 Weak sequential compactness	p. 302
15 The weak* topology	p. 303
16 The Alaoglu theorem	p. 304
17 Hilbert spaces	p. 306
17.1 The Schwarz inequality	p. 307
17.2 The parallelogram identity	p. 307
18 Orthogonal sets, representations, and functionals	p. 308
18.1 Bounded linear functionals on H	p. 310
19 Orthonormal systems	p. 310
19.1 The Bessel inequality	p. 311
19.2 Separable Hilbert spaces	p. 312
20 Complete orthonormal systems	p. 312
20.1 Equivalent notions of complete systems	p. 313
20.2 Maximal and complete orthonormal systems	p. 313
20.3 The Gram-Schmidt orthonormalization process	p. 314
20.4 On the dimension of a separable Hilbert space	p. 314
Problems and Complements	p. 314
VII Spaces of Continuous Functions, Distributions, and Weak Derivatives	p. 325
1 Spaces of continuous functions	p. 325
1.1 Partition of unity	p. 326
2 Bounded linear functionals on C[subscript o](R[superscript N])	p. 327
2.1 Remarks on functionals of the type (2.2) and (2.3)	p. 327
2.2 Characterizing C[subscript o](R[superscript N])	p. 328
3 Positive linear functionals on C[subscript o](R[superscript N])	p. 328
4 Proof of Theorem 3.3: Constructing the measure [mu]	p. 331
5 Proof of Theorem 3.3: Representing T as in (3.3)	p. 333
6 Characterizing bounded linear functionals on C[subscript o](R[superscript N])	p. 335
6.1 Locally bounded linear functionals on C[subscript o](R[superscript N])	p. 335
6.2 Bounded linear functionals on C[subscript o](R[superscript N])	p. 336
7 A topology for C[superscript infinity subscript o] (E) for an open set E [subset or is implied by] R[superscript N]	p. 337
8 A metric topology for C[superscript infinity subscript o] (E)	p. 339
8.1 Equivalence of these topologies	p. 340
8.2 D(E) is not complete	p. 341
9 A topology for C[superscript infinity subscript o] (K) for a compact set K [subset or is implied by] E	p. 341
9.1 A metric topology for C[superscript infinity subscript o] (K)	p. 342
9.2 D(K) is complete	p. 342
10 Relating the topology of D(E) to the topology of D(K)	p. 343
10.1 Noncompleteness of D(E)	p. 344
11 The Schwartz topology of D(E)	p. 344
12 D(E) is complete	p. 346
12.1 Cauchy sequences in D(E)	p. 347
12.2 The topology of D(E) is not metrizable	p. 347
13 Continuous maps and functionals	p. 348
13.1 Distributions on E	p. 348
13.2 Continuous linear maps T: D(E) [right arrow] D(E)	p. 349
14 Distributional derivatives	p. 349
14.1 Derivatives of distributions	p. 350
14.2 Some examples	p. 350
14.3 Miscellaneous remarks	p. 351
15 Fundamental Solutions	p. 352
15.1 The fundamental solution of the wave operator	p. 352
15.2 The fundamental solution of the Laplace operator	p. 354
16 Weak derivatives and main properties	p. 355
17 Domains and their boundaries	p. 358
17.1 [partial differentialE of class C[superscript 1]	p. 358
17.2 Positive geometric density	p. 358
17.3 The segment property	p. 358
17.4 The cone property	p. 359
17.5 On the various properties of [partial differential] E	p. 359
18 More on smooth approximations	p. 359
19 Extensions into R[superscript N]	p. 361
20 The chain rule	p. 363
21 Steklov averagings	p. 365
22 Characterizing W[superscript 1,p] (E) for 1	p. 367
22.1 Remarks on W[superscript 1, infinity] (E)	p. 368
23 The Rademacher theorem	p. 368
Problems and Complements	p. 371
VIII Topics on Integrable Functions of Real Variables	p. 375
1 Vitali-type coverings	p. 375
2 The maximal function	p. 377
3 Strong L[superscript p] estimates for the maximal function	p. 379
3.1 Estimates of weak and strong type	p. 380
4 The Calderon-Zygmund decomposition theorem	p. 381
5 Functions of bounded mean oscillation	p. 383
6 Proof of Theorem 5.1	p. 384
7 The sharp maximal function	p. 387
8 Proof of the Fefferman-Stein theorem	p. 388
9 The Marcinkiewicz interpolation theorem	p. 390
9.1 Quasi-linear maps and interpolation	p. 391
10 Proof of the Marcinkiewicz theorem	p. 392
11 Rearranging the values of a function	p. 394
12 Basic properties of rearrangements	p. 396
13 Symmetric rearrangements	p. 398
14 A convolution inequality for rearrangements	p. 400
14.1 Approximations by simple functions	p. 400
15 Reduction to a finite union of intervals	p. 402
16 Proof of Theorem 14.1: The case where T + S [less than or equal] R	p. 404
17 Proof of Theorem 14.1: The case where S + T > R	p. 404
17.1 Proof of Lemma 17.1	p. 407
18 Hardy's inequality	p. 407
19 A convolution-type inequality	p. 409
19.1 Some reductions	p. 409
20 Proof of Theorem 19.1	p. 410
21 An equivalent form of Theorem 19.1	p. 411
22 An N-dimensional version of Theorem 21.1	p. 412
23 L[superscript p] estimates of Riesz potentials	p. 413
24 The limiting case p = N	p. 415
Problems and Complements	p. 417
IX Embeddings of W[superscript 1,p] (E) into L[superscript q] (E)	p. 423
1 Multiplicative embeddings of W[subscript o superscript 1,p] (E)	p. 423
2 Proof of Theorem 1.1 for N = 1	p. 425
3 Proof of Theorem 1.1 for 1 [less than or equal] p	p. 425
4 Proof of Theorem 1.1 for 1 [less than or equal] p	p. 428
5 Proof of Theorem 1.1 for p [greater than or equal] N > 1	p. 428
5.1 Estimate of I[subscript 1](x, R)	p. 429
5.2 Estimate of I[subscript 2](x, R)	p. 430
6 Proof of Theorem 1.1 for p [greater than or equal] N > 1, concluded	p. 430
7 On the limiting case p = N	p. 431
8 Embeddings of W[superscript 1,p] (E)	p. 432
9 Proof of Theorem 8.1	p. 433
10 Poincare inequalities	p. 435
10.1 The Poincare inequality	p. 435
10.2 Multiplicative Poincare inequalities	p. 437
11 The discrete isoperimetric inequality	p. 438
12 Morrey spaces	p. 439
12.1 Embeddings for functions in the Morrey spaces	p. 440
13 Limiting embedding of W[superscript 1,N] (E)	p. 441
14 Compact embeddings	p. 443
15 Fractional Sobolev spaces in R[superscript N]	p. 445
16 Traces	p. 447
17 Traces and fractional Sobolev spaces	p. 448
18 Traces on [partial differential]E of functions in W[superscript 1,p] (E)	p. 450
18.1 Traces and fractional Sobolev spaces	p. 453
19 Multiplicative embeddings of W[superscript 1,p] (E)	p. 453
20 Proof of Theorem 19.1: A special case	p. 456
21 Constructing a map between E and Q	p. 458
22 Constructing a map between E and Q: Part 2	p. 460
23 Proof of Theorem 19.1, concluded	p. 463
Problems and Complements	p. 464
References	p. 469
Index	p. 473

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