1 Measure and integral 4
1.1 Measure 4
1.2 Measurable functions 7
1.3 Integration 9
1.4 Notes and remarks 12
2 The Cauchy-Schwarz inequality 13
2.1 Cauchy's inequality 13
2.2 Inner-product spaces 14
2.3 The Cauchy-Schwarz inequality 15
2.4 Notes and remarks 17
3 The AM-GM inequality 19
3.1 The AM GM inequality 19
3.2 Applications 21
3.3 Notes and remarks 23
4 Convexity,and Jensen's inequality 24
4.1 Convex sets and convex functions 24
4.2 Convex functions on an interval 26
4.3 Directional derivatives and sublinear functionals 29
4.4 The Hahn-Banach theorem 31
4.5 Normed spaces,Banach spaces and Hilbert space 34
4.6 The Hahn-Banach theorem for normed spaces 36
4.7 Barycentres and weak integrals 39
4.8 Notes and remarks 40
5 The Lp spaces 45
5.1 Lp spaces,and Minkowski's inequality 45
5.2 The Lebesgue decomposition theorem 47
5.3 The reverse Minkowski inequality 49
5.4 H？lder's inequality 50
5.5 The inequalities of Liapounov and Littlewood 54
5.6 Duality 55
5.7 The Loomis-Whitney inequality 57
5.8 A Sobolev inequality 60
5.9 Schur's theorem and Schur's test 62
5.10 Hilbert's absolute inequality 65
5.11 Notes and remarks 67
6 Banach function spaces 70
6.1 Banach function spaces 70
6.2 Function space duality 72
6.3 Orlicz spaces 73
6.4 Notes and remarks 76
7 Rearrangements 78
7.1 Decreasing rearrangements 78
7.2 Rearrangement-invariant Banach function spaces 80
7.3 Muirhead's maximal function 81
7.4 Majorization 84
7.5 Calderón's interpolation theorem and its converse 88
7.6 Symmetric Banach sequence spaces 91
7.7 The method of transference 93
7.8 Finite doubly stochastic matrices 97
7.9 Schur convexity 98
7.10 Notes and remarks 100
8 Maximal inequalities 103
8.1 The Hardy-Riesz inequality（1＜p＜∞） 103
8.2 The Hardy-Riesz inequality(p=1) 105
8.3 Related inequalities 106
8.4 Strong type and weak type 108
8.5 Riesz weak type 111
8.6 Hardy,Littlewood,and a batsman's averages 112
8.7 Riesz's sunrise lemma 114
8.8 Differentiation almost everywhere 117
8.9 Maximal operators in higher dimensions 118
8.10 The Lebesgue density theorem 121
8.11 Convolution kernels 121
8.12 Hedberg's inequality 125
8.13 Martingales 127
8.14 Doob's inequality 130
8.15 The martingale convergence theorem 130
8.16 Notes and remarks 133
9 Complex interpolation 135
9.1 Hadamard's three lines inequality 135
9.2 Compatible couples and intermediate spaces 136
9.3 The Riesz—Thorin interpolation theorem 138
9.4 Young's inequality 140
9.5 The Hausdorff—Young inequality 141
9.6 Fourier type 143
9.7 The generalized Clarkson inequalities 145
9.8 Uniform convexity 147
9.9 Notes and remarks 150
10 Real interpolation 154
10.1 The Marcinkiewicz interpolation theorem：Ⅰ 154
10.2 Lorentz spaces 156
10.3 Hardy's inequality 158
10.4 The scale of Lorentz spaces 159
10.5 The Marcinkiewicz interpolation theorem：Ⅱ 162
10.6 Notes and remarks 165
11 The Hilbert transform,and Hilbert's inequalities 167
11.1 The conjugate Poisson kernel 167
11.2 The Hilbert transform on L2(R) 168
11.3 The Hilbert transform on Lp(R)for 1＜p＜∞ 170
11.4 Hilbert's inequality for sequences 174
11.5 The Hilbert transform on T 175
11.6 Multipliers 179
11.7 Singular integral operators 180
11.8 Singular integral operators on Lp(Rd)for 1≤p＜∞ 183
11.9 Notes and remarks 185
12 Khintchine's inequality 187
12.1 The contraction principle 187
12.2 The reflection principle,and Lévy's inequalities 189
12.3 Khintchine's inequality 192
12.4 The law of the iterated logarithm 194
12.5 Strongly embedded subspaces 196
12.6 Stable random variables 198
12.7 Sub-Gaussian random variables 199
12.8 Kahane's theorem and Kahane's inequality 201
12.9 Notes and remarks 204
13 Hypercontractive and logarithmic Sobolev inequalities 206
13.1 Bonami's inequality 206
13.2 Kahane's inequality revisited 210
13.3 The theorem of Latala and Oleszkiewicz 211
13.4 The logarithmic Sobolev inequality on Dd 2 213
13.5 Gaussian measure and the Hermite polynomials 216
13.6 The central limit theorem 219
13.7 The Gaussian hypercontractive inequality 221
13.8 Correlated Gaussian random variables 223
13.9 The Gaussian logarithmic Sobolev inequality 225
13.10 The logarithmic Sobolev inequality in higher dimensions 227
13.11 Beckner's inequality 229
13.12 The Babenko-Beckner inequality 230
13.13 Notes and remarks 232
14 Hadamard's inequality 233
14.1 Hadamard's inequality 233
14.2 Hadamard numbers 234
14.3 Error-correcting codes 237
14.4 Note and remark 238
15 Hilbert space operator inequalities 239
15.1 Jordan normal form 239
15.2 Riesz operators 240
15.3 Related operators 241
15.4 Compact operators 242
15.5 Positive compact operators 243
15.6 Compact operators between Hilbert spaces 245
15.7 Singular numbers,and the Rayleigh-Ritz minimax formula 246
15.8 Weyl's inequality and Horn's inequality 247
15.9 Ky Fan's inequality 250
15.10 Operator ideals 251
15.11 The Hilbert-Schmidt class 253
15.12 The trace class 256
15.13 Lidskii's trace formula 257
15.14 Operator ideal duality 260
15.15 Notes and remarks 261
16 Summing operators 263
16.1 Unconditional convergence 263
16.2 Absolutely summing operators 265
16.3 (p,q)-summing operators 266
16.4 Examples of p-summing operators 269
16.5 (p,2)-summing operators between Hilbert spaces 271
16.6 Positive operators on L1 273
16.7 Mercer's theorem 274
16.8 p-summing operators between Hilbert spaces(1≤p≤2) 276
16.9 Pietsch's domination theorem 277
16.10 Pietsch's factorization theorem 279
16.11 p-summing operators between Hilbert spaces（2≤p ≤∞） 281
16.12 The Dvoretzky-Rogers theorem 282
16.13 Operators that factor through a Hilbert space 284
16.14 Notes and remarks 287
17 Approximation numbers and eigenvalues 289
17.1 The approximation,Gelfand and Weyl numbers 289
17.2 Subadditive and submultiplicative properties 291
17.3 Pietsch's inequality 294
17.4 Eigenvalues of p-summing and(p,2)-summing endomorphisms 296
17.5 Notes and remarks 301
18 Grothendieck's inequality,type and cotype 302
18.1 Littlewood's 4/3 inequality 302
18.2 Grothendieck's inequality 304
18.3 Grothendieck's theorem 306
18.4 Another proof,using Paley's inequality 307
18.5 The little Grothendieck theorem 310
18.6 Type and cotype 312
18.7 Gaussian type and cotype 314
18.8 Type and cotype of Lp spaces 316
18.9 The little Grothendieck theorem revisited 318
18.10 More on cotype 320
18.11 Notes and remarks 323