Modern analysis

Title Page......Page 2
Copyright Information......Page 3
Preface......Page 4
Contents......Page 6
1 Set theory......Page 12
2 General topology......Page 15
3 Urysohn\'s lemma......Page 20
4 Exercises......Page 23
1 Compactness in metric space......Page 28
2 Compactness in spaces of continuous functions......Page 32
3 Stone Weierstrass theorem......Page 35
4 Exercises......Page 40
1 Baire category theorem......Page 44
2 Uniform boundedness closed graph and open mapping theorems......Page 47
3 Hahn Banach theorem......Page 51
4 Exercises......Page 56
4 Hilbert Spaces......Page 60
1 Finite dimensional normed linear space......Page 64
2 Uniformly convex Banach spaces......Page 66
3 Exercises......Page 71
1 The derivative......Page 74
2 Finite dimensions......Page 76
3 Higher order derivatives......Page 79
4 Inverse function theorem......Page 81
5 Ordinary differential equations......Page 89
6 Exercises......Page 91
6 Locally Convex Topological Vector Spaces......Page 98
1 Separation theorems......Page 102
2 The weak and weak* topologies......Page 108
3 The Tychonoff fixed point theorem......Page 116
4 Set-valued maps......Page 121
5 Finite dimensional spaces......Page 126
6 Exercises......Page 130
2 Monotone classes and algebras......Page 134
3 Exercises......Page 143
8 The Abstract Lebesgue Integral......Page 146
1 The space L^1......Page 150
2 Double sums of nonnegative terms......Page 155
3 Exercises......Page 156
1 Outer measures......Page 160
2 Regular measures......Page 165
4 Exercises......Page 179
1 Lebesgue measure......Page 182
2 Iterated integrals......Page 186
3 Change of variables......Page 188
4 Polar coordinates......Page 194
5 Exercises......Page 197
11 Product Measure......Page 200
1 Completion of product measure......Page 206
2 Exercises......Page 210
1 Basic inequalities and properties......Page 214
2 Density of simple functions......Page 218
3 Continuity of translation......Page 220
4 Separability......Page 221
5 Mollifiers and density of smooth functions......Page 222
6 Exercises......Page 224
1 Radon Nikodym Theorem......Page 230
2 Vector measures......Page 232
3 Representation theorems for the dual space of L^p the er finite case......Page 237
4 Riesz Representation theorem for non er finite measure spaces......Page 243
5 The dual space of C(X)......Page 248
6 Exercises......Page 252
1 The Vitali covering theorem......Page 256
2 Differentiation with respect to Lebesgue measure......Page 258
3 The change of variables theorem for multiple integrals......Page 262
4 Exercises......Page 270
1 Besicovitch covering theorem......Page 274
2 Differentiation with respect to Radon measures......Page 279
3 Slicing measures......Page 282
4 Young measures......Page 288
5 Exercises......Page 292
1 The Schwartz class......Page 294
2 Fourier transforms of functions in L^2(R^n)......Page 299
3 Tempered distributions......Page 304
4 Exercises......Page 310
1 Random vectors......Page 314
2 Conditional probability and independence......Page 319
3 Conditional expectation......Page 327
4 Conditional expectation given a σ algebra......Page 331
5 Strong law of large numbers......Page 341
6 The normal distribution......Page 346
7 The central limit theorem......Page 349
8 The continuity theorem......Page 353
9 Exercises......Page 359
1 Test functions and weak derivatives......Page 366
2 Weak derivatives in L^p_loc......Page 370
3 Morrey\'s inequality......Page 372
4 Rademacher\'s theorem......Page 374
5 Exercises......Page 377
19 Hausdorff Measures......Page 382
1 Steiner symmetrization......Page 384
2 The isodiametric inequality......Page 386
3 Hausdorff measures......Page 388
4 Properties of Hausdorff measure......Page 389
1 Lipschitz mappings......Page 398
2 The area formula for one to one Lipschitz mappings......Page 406
3 Mappings that are not one to one......Page 409
4 Surface measure......Page 413
5 The divergence theorem......Page 416
6 Exercises......Page 423
1 A determinant identity......Page 430
2 The Coarea formula......Page 432
3 Change of variables......Page 443
4 Exercises......Page 444
1 The Marcinkiewicz interpolation theorem......Page 446
2 The Calderon Zygmund decomposition......Page 449
3 Mihlin\'s theorem......Page 451
4 Singular integrals......Page 464
5 The Helmholtz decomposition of vector fields......Page 473
6 Exercises......Page 480
1 Strong and weak measurability......Page 486
2 The Bochner integral......Page 492
3 Measurable representatives......Page 499
4 Vector measures......Page 501
5 The Riesz representation theorem......Page 505
6 Exercises......Page 510
1 Continuity properties of convex functions......Page 514
2 Separation properties......Page 517
3 Conjugate functions......Page 520
4 Subgradients......Page 522
5 Hilbert space......Page 530
6 Exercises......Page 535
Appendix 1: The Hausdorff Maximal theorem......Page 540
1 Exercises......Page 543
Appendix 2: Stone\'s Theorem and Partitions of Unity......Page 546
1 General partitions of unity......Page 551
2 A general metrization theorem......Page 552
1 Taylor\'s formula......Page 556
2 Analytic functions......Page 557
3 Ordinary differential equations......Page 563
Appendix 4: The Brouwer Fixed Point theorem......Page 568
References......Page 572
Index......Page 580