Title ......Page 1
Contents ......Page 3
Preface ......Page 12
Acknowledgments ......Page 20
1 Countable sets ......Page 22
2 The Cantor set ......Page 23
3 Cardinality ......Page 25
3.1 Some examples ......Page 26
4 Cardinality of some infinite Cartesian products ......Page 27
5 Orderings, the maximal principle, and the axiom of choice ......Page 29
6 Well-ordering ......Page 30
Problems and Complements ......Page 32
1 Topological spaces ......Page 38
2 Urysohn's lemma ......Page 40
3 The Tietze extension theorem ......Page 42
4 Bases, axioms of countability, and product topologies ......Page 43
4.1 Product topologies ......Page 45
5 Compact topological spaces ......Page 46
5.1 Sequentially compact topological spaces ......Page 47
6 Compact subsets of RN ......Page 48
7 Continuous functions on countably compact spaces ......Page 50
8 Products of compact spaces ......Page 51
9 Vector spaces ......Page 52
9.2 Linear maps and isomorphisms ......Page 54
10 Topological vector spaces ......Page 55
10.1 Boundedness and continuity ......Page 56
12 Finite-dimensional topological vector spaces ......Page 57
12.1 Locally compact spaces ......Page 58
13 Metric spaces ......Page 59
13.1 Separation and axioms of countability ......Page 60
13.3 Pseudometrics ......Page 61
14 Metric vector spaces ......Page 62
14.1 Maps between metric spaces ......Page 63
15 Spaces of continuous functions ......Page 64
16 On the structure of a complete metric space ......Page 65
17 Compact and totally bounded metric spaces ......Page 67
17.1 Precompact subsets of X ......Page 69
Problems and Complements ......Page 70
1 Partitioning open subsets of RN ......Page 86
2 Limits of sets, characteristic functions, and si-algebras ......Page 88
3 Measures ......Page 89
3.2 Some examples ......Page 92
4 Outer measures and sequential coverings ......Page 93
4.2 The Lebesgue-Stieltjes outer measure ......Page 94
5 The Hausdorff outer measure in RN ......Page 95
6 Constructing measures from outer measures ......Page 97
7 The Lebesgue-Stieltjes measure on R ......Page 100
8 The Hausdorff measure on RN ......Page 101
9 Extending measures from semialgebras to si-algebras ......Page 103
10 Necessary and sufficient conditions for measurability ......Page 105
11 More on extensions from semialgebras to si-algebras ......Page 107
12.1 A necessary and sufficient condition of measurability ......Page 109
13 A nonmeasurable set ......Page 111
14.1 A continuous increasing function f:[0,1]->[0,1] ......Page 112
14.2 On the preimage of a measurable set ......Page 114
15 More on Borel measures ......Page 115
15.2 Regular Borel measures and Radon measures ......Page 118
16 Regular outer measures and Radon measures ......Page 119
17 Vitali coverings ......Page 120
18 The Besicovitch covering theorem ......Page 124
19 Proof of Proposition 18.2 ......Page 126
20 The Besicovitch measure-theoretical covering theorem ......Page 128
Problems and Complements ......Page 131
1 Measurable functions ......Page 144
2 The Egorov theorem ......Page 147
3 Approximating measurable functions by simple functions ......Page 149
4 Convergence in measure ......Page 151
5 Quasi-continuous functions and Lusin's theorem ......Page 154
6 Integral of simple functions ......Page 156
7 The Lebesgue integral of nonnegative functions ......Page 157
8 Fatou's lemma and the monotone convergence theorem ......Page 158
9 Basic properties of the Lebesgue integral ......Page 160
10 Convergence theorems ......Page 162
12 Product of measures ......Page 163
13 On the structure of (A*B) ......Page 165
14 The Fubini-Tonelli theorem ......Page 168
15.1 Integrals in terms of distribution functions ......Page 169
15.2 Convolution integrals ......Page 170
15.3 The Marcinkiewicz integral ......Page 171
16 Signed measures and the Hahn decomposition ......Page 172
17 The Radon-Nikodym theorem ......Page 175
18.1 The Jordan decomposition ......Page 178
18.2 The Lebesgue decomposition ......Page 180
Problems and Complements ......Page 181
1 Functions of bounded variations ......Page 192
2 Dini derivatives ......Page 194
3 Differentiating functions of bounded variation ......Page 197
4 Differentiating series of monotone functions ......Page 198
5 Absolutely continuous functions ......Page 200
6 Density of a measurable set ......Page 202
7 Derivatives of integrals ......Page 203
8 Differentiating Radon measures ......Page 205
9 Existence and measurability of D ......Page 207
9.1 Proof of Proposition 9.2 ......Page 209
10.1 Representing D for < ......Page 210
11 The Lebesgue differentiation theorem ......Page 212
11.2 Lebesgue points of an integrable function ......Page 213
12 Regular families ......Page 214
13 Convex functions ......Page 215
14 Jensen's inequality ......Page 217
15 Extending continuous functions ......Page 218
16 The Weierstrass approximation theorem ......Page 220
17 The Stone-Weierstrass theorem ......Page 221
18 Proof of the Stone-Weierstrass theorem ......Page 222
18.1 Proof of Stone's theorem ......Page 223
19 The Ascoli-Arzete theorem ......Page 224
19.1 Precompact subsets of N(E) ......Page 225
Problems and Complements ......Page 226
1 Functions in LP(E) and their norms ......Page 242
1.2 The spaces Lq for q<0 ......Page 243
2 The Holder and Minkowski inequalities ......Page 244
3 The reverse Holder and Minkowski inequalities ......Page 245
4.1 Characterizing the norm ||f||p for 1
4.2 The norm ||.||oo for E of finite measure ......Page 247
5 LP(E) for 15.1 LP(E) for 16.1 Open convex subsets of LP(E) when 07 Convergence in LP{E) and completeness ......Page 251
8 Separating LP(E) by simple functions ......Page 253
9.1 A counterexample ......Page 255
10 Weak lower semicontinuity of the norm in LP(E) ......Page 256
11 Weak convergence and norm convergence ......Page 257
11.2 Proof of Proposition 11.1 for 1 < p < 2 ......Page 258
12 Linear functionals in LP{E) ......Page 259
13 The Riesz representation theorem ......Page 260
13.1 Proof of Theorem 13.1 :The case where {X,A,mu) is finite ......Page 261
13.2 Proof of Theorem 13.1: The case where {X,A,mu} is si-finite ......Page 262
13.3 Proof of Theorem 13.1: The case where 114 The Hanner and Clarkson inequalities ......Page 264
14.1 Proof of Hanner's inequalities ......Page 265
14.2 Proof of Clarkson's inequalities ......Page 266
15 Uniform convexity of LP(E) for 116.1 Proof of Theorem 13.1: The case where 116.2 The case where p=1 and E is of finite measure ......Page 269
16.3 The case where p=1 and {X,A,mu} is si-finite ......Page 270
17.1 An alternate proof of Proposition 17.1 ......Page 271
18 If E subset RN and p\in [1,oo), then LP(E) is separable ......Page 272
19 Selecting weakly convergent subsequences ......Page 275
20 Continuity of the translation in LP{E) for 121 Approximating functions in LP(E) with functions in C8(E) ......Page 278
22 Characterizing precompact sets in LP(E) ......Page 281
Problems and Complements ......Page 283
1 Normed spaces ......Page 296
1.1 Seminorms and quotients ......Page 297
2.1 A counterexample ......Page 298
2.2 The Riesz lemma ......Page 299
2.3 Finite-dimensional spaces ......Page 300
3 Linear maps and functionals ......Page 301
4 Examples of maps and functionals ......Page 303
4.2 Linear functionals on C(E) ......Page 304
5 Kernels of maps and functionals ......Page 305
6 Equibounded families of linear maps ......Page 306
7 Contraction mappings ......Page 307
7.1 Applications to some Fredholm integral equations ......Page 308
8 The open mapping theorem ......Page 309
8.2 The closed graph theorem ......Page 310
9 The Hahn-Banach theorem ......Page 311
10 Some consequences of the Hahn-Banach theorem ......Page 313
11 Separating convex subsets of X ......Page 316
12 Weak topologies ......Page 318
12.1 Weakly and strongly closed convex sets ......Page 320
13 Reflexive Banach spaces ......Page 321
14 Weak compactness ......Page 322
14.1 Weak sequential compactness ......Page 323
15 The weak* topology ......Page 324
16 The Alaoglu theorem ......Page 325
17 Hubert spaces ......Page 327
17.2 The parallelogram identity ......Page 328
18 Orthogonal sets, representations, and functionals ......Page 329
19 Orthonormal systems ......Page 331
19.1 The Bessel inequality ......Page 332
20 Complete orthonormal systems ......Page 333
20.2 Maximal and complete orthonormal systems ......Page 334
Problems and Complements ......Page 335
1 Spaces of continuous functions ......Page 346
1.1 Partition of unity ......Page 347
2.1 Remarks on functionals of the type (2.2) and (2.3) ......Page 348
3 Positive linear functionals on C0(RN) ......Page 349
4 Proof of Theorem 3.3: Constructing the measure mu ......Page 352
5 Proof of Theorem 3.3: Representing T as in (3.3) ......Page 354
6.1 Locally bounded linear functionals on C0(RN) ......Page 356
6.2 Bounded linear functionals on C0(RN) ......Page 357
7 A topology for C_08(E) for an open set E N RN ......Page 358
8 A metric topology for C8(E) ......Page 360
8.1 Equivalence of these topologies ......Page 361
9 A topology for C8(K) for a compact set E N E ......Page 362
9.2 D(K) is complete ......Page 363
10 Relating the topology of D(E) to the topology of D(K) ......Page 364
11 The Schwartz topology of D(E) ......Page 365
12 D(E) is complete ......Page 367
12.2 The topology of D(E) is not metrizable ......Page 368
13.1 Distributions on ? ......Page 369
14 Distributional derivatives ......Page 370
14.2 Some examples ......Page 371
14.3 Miscellaneous remarks ......Page 372
15.1 The fundamental solution of the wave operator ......Page 373
15.2 The fundamental solution of the Laplace operator ......Page 375
16 Weak derivatives and main properties ......Page 376
17.3 The segment property ......Page 379
18 More on smooth approximations ......Page 380
19 Extensions into RN ......Page 382
20 The chain rule ......Page 384
21 Steklov averagings ......Page 386
22 Characterizing Wl>p(E) for 1 23 The Rademacher theorem ......Page 389
Problems and Complements ......Page 392
1 Vitali-type coverings ......Page 396
2 The maximal function ......Page 398
3 Strong Lp estimates for the maximal function ......Page 400
3.1 Estimates of weak and strong type ......Page 401
4 The Calderon-Zygmund decomposition theorem ......Page 402
5 Functions of bounded mean oscillation ......Page 404
6 Proof of Theorem 5.1 ......Page 405
7 The sharp maximal function ......Page 408
8 Proof of the Fefferman-Stein theorem ......Page 409
9 The Marcinkiewicz interpolation theorem ......Page 411
9.1 Quasi-linear maps and interpolation ......Page 412
10 Proof of the Marcinkiewicz theorem ......Page 413
11 Rearranging the values of a function ......Page 415
12 Basic properties of rearrangements ......Page 417
13 Symmetric rearrangements ......Page 419
14.1 Approximations by simple functions ......Page 421
15 Reduction to a finite union of intervals ......Page 423
17 Proof of Theorem 14.1: The case where S+T>R ......Page 425
18 Hardy's inequality ......Page 428
19.1 Some reductions ......Page 430
20 Proof of Theorem 19.1 ......Page 431
21 An equivalent form of Theorem 19.1 ......Page 432
22 An'N-dimensional version of Theorem 21.1 ......Page 433
23 Lp estimates of Riesz potentials ......Page 434
24 The limiting case p=N ......Page 436
Problems and Complements ......Page 438
1 Multiplicative embeddings of W1p(E) ......Page 444
3 Proof of Theorem 1.1 for 15 Proof of Theorem 1.1 for p>N>1 ......Page 449
5.1 Estimate of I1(x,R) ......Page 450
6 Proof of Theorem 1.1 for p>N>1, concluded ......Page 451
7 On the limiting case p=N ......Page 452
8 Embeddings of Wlp(E) ......Page 453
9 Proof of Theorem 8.1 ......Page 454
10.1 The Poincare inequality ......Page 456
10.2 Multiplicative Poincare inequalities ......Page 458
11 The discrete isoperimetric inequality ......Page 459
12 Morrey spaces ......Page 460
12.1 Embeddings for functions in the Morrey spaces ......Page 461
13 Limiting embedding of W1N(E) ......Page 462
14 Compact embeddings ......Page 464
15 Fractional Sobolev spaces in RN ......Page 466
16 Traces ......Page 468
17 Traces and fractional Sobolev spaces ......Page 469
18 Traces on dE of functions in W1p(E) ......Page 471
19 Multiplicative embeddings of W1p(E) ......Page 474
20 Proof of Theorem 19.1: A special case ......Page 477
21 Constructing a map between E and Q: Part 1 ......Page 479
22 Constructing a map between E and Q: Part 2 ......Page 481
23 Proof of Theorem 19.1, concluded ......Page 484
Problems and Complements ......Page 485
References ......Page 490
Index ......Page 494
