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1 Measure and integral4

1.1 Measure4

1.2 Measurable functions7

1.3 Integration9

1.4 Notes and remarks12

2 The Cauchy-Schwarz inequality13

2.1 Cauchy's inequality13

2.2 Inner-product spaces14

2.3 The Cauchy-Schwarz inequality15

2.4 Notes and remarks17

3 The AM-GM inequality19

3.1 The AM GM inequality19

3.2 Applications21

3.3 Notes and remarks23

4 Convexity,and Jensen's inequality24

4.1 Convex sets and convex functions24

4.2 Convex functions on an interval26

4.3 Directional derivatives and sublinear functionals29

4.4 The Hahn-Banach theorem31

4.5 Normed spaces,Banach spaces and Hilbert space34

4.6 The Hahn-Banach theorem for normed spaces36

4.7 Barycentres and weak integrals39

4.8 Notes and remarks40

5 The Lp spaces45

5.1 Lp spaces,and Minkowski's inequality45

5.2 The Lebesgue decomposition theorem47

5.3 The reverse Minkowski inequality49

5.4 H？lder's inequality50

5.5 The inequalities of Liapounov and Littlewood54

5.6 Duality55

5.7 The Loomis-Whitney inequality57

5.8 A Sobolev inequality60

5.9 Schur's theorem and Schur's test62

5.10 Hilbert's absolute inequality65

5.11 Notes and remarks67

6 Banach function spaces70

6.1 Banach function spaces70

6.2 Function space duality72

6.3 Orlicz spaces73

6.4 Notes and remarks76

7 Rearrangements78

7.1 Decreasing rearrangements78

7.2 Rearrangement-invariant Banach function spaces80

7.3 Muirhead's maximal function81

7.4 Majorization84

7.5 Calderón's interpolation theorem and its converse88

7.6 Symmetric Banach sequence spaces91

7.7 The method of transference93

7.8 Finite doubly stochastic matrices97

7.9 Schur convexity98

7.10 Notes and remarks100

8 Maximal inequalities103

8.1 The Hardy-Riesz inequality（1＜p＜∞）103

8.2 The Hardy-Riesz inequality(p=1)105

8.3 Related inequalities106

8.4 Strong type and weak type108

8.5 Riesz weak type111

8.6 Hardy,Littlewood,and a batsman's averages112

8.7 Riesz's sunrise lemma114

8.8 Differentiation almost everywhere117

8.9 Maximal operators in higher dimensions118

8.10 The Lebesgue density theorem121

8.11 Convolution kernels121

8.12 Hedberg's inequality125

8.13 Martingales127

8.14 Doob's inequality130

8.15 The martingale convergence theorem130

8.16 Notes and remarks133

9 Complex interpolation135

9.1 Hadamard's three lines inequality135

9.2 Compatible couples and intermediate spaces136

9.3 The Riesz—Thorin interpolation theorem138

9.4 Young's inequality140

9.5 The Hausdorff—Young inequality141

9.6 Fourier type143

9.7 The generalized Clarkson inequalities145

9.8 Uniform convexity147

9.9 Notes and remarks150

10 Real interpolation154

10.1 The Marcinkiewicz interpolation theorem：Ⅰ154

10.2 Lorentz spaces156

10.3 Hardy's inequality158

10.4 The scale of Lorentz spaces159

10.5 The Marcinkiewicz interpolation theorem：Ⅱ162

10.6 Notes and remarks165

11 The Hilbert transform,and Hilbert's inequalities167

11.1 The conjugate Poisson kernel167

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 sequences174

11.5 The Hilbert transform on T175

11.6 Multipliers179

11.7 Singular integral operators180

11.8 Singular integral operators on Lp(Rd)for 1≤p＜∞183

11.9 Notes and remarks185

12 Khintchine's inequality187

12.1 The contraction principle187

12.2 The reflection principle,and Lévy's inequalities189

12.3 Khintchine's inequality192

12.4 The law of the iterated logarithm194

12.5 Strongly embedded subspaces196

12.6 Stable random variables198

12.7 Sub-Gaussian random variables199

12.8 Kahane's theorem and Kahane's inequality201

12.9 Notes and remarks204

13 Hypercontractive and logarithmic Sobolev inequalities206

13.1 Bonami's inequality206

13.2 Kahane's inequality revisited210

13.3 The theorem of Latala and Oleszkiewicz211

13.4 The logarithmic Sobolev inequality on Dd 2213

13.5 Gaussian measure and the Hermite polynomials216

13.6 The central limit theorem219

13.7 The Gaussian hypercontractive inequality221

13.8 Correlated Gaussian random variables223

13.9 The Gaussian logarithmic Sobolev inequality225

13.10 The logarithmic Sobolev inequality in higher dimensions227

13.11 Beckner's inequality229

13.12 The Babenko-Beckner inequality230

13.13 Notes and remarks232

14 Hadamard's inequality233

14.1 Hadamard's inequality233

14.2 Hadamard numbers234

14.3 Error-correcting codes237

14.4 Note and remark238

15 Hilbert space operator inequalities239

15.1 Jordan normal form239

15.2 Riesz operators240

15.3 Related operators241

15.4 Compact operators242

15.5 Positive compact operators243

15.6 Compact operators between Hilbert spaces245

15.7 Singular numbers,and the Rayleigh-Ritz minimax formula246

15.8 Weyl's inequality and Horn's inequality247

15.9 Ky Fan's inequality250

15.10 Operator ideals251

15.11 The Hilbert-Schmidt class253

15.12 The trace class256

15.13 Lidskii's trace formula257

15.14 Operator ideal duality260

15.15 Notes and remarks261

16 Summing operators263

16.1 Unconditional convergence263

16.2 Absolutely summing operators265

16.3 (p,q)-summing operators266

16.4 Examples of p-summing operators269

16.5 (p,2)-summing operators between Hilbert spaces271

16.6 Positive operators on L1273

16.7 Mercer's theorem274

16.8 p-summing operators between Hilbert spaces(1≤p≤2)276

16.9 Pietsch's domination theorem277

16.10 Pietsch's factorization theorem279

16.11 p-summing operators between Hilbert spaces（2≤p ≤∞）281

16.12 The Dvoretzky-Rogers theorem282

16.13 Operators that factor through a Hilbert space284

16.14 Notes and remarks287

17 Approximation numbers and eigenvalues289

17.1 The approximation,Gelfand and Weyl numbers289

17.2 Subadditive and submultiplicative properties291

17.3 Pietsch's inequality294

17.4 Eigenvalues of p-summing and(p,2)-summing endomorphisms296

17.5 Notes and remarks301

18 Grothendieck's inequality,type and cotype302

18.1 Littlewood's 4/3 inequality302

18.2 Grothendieck's inequality304

18.3 Grothendieck's theorem306

18.4 Another proof,using Paley's inequality307

18.5 The little Grothendieck theorem310

18.6 Type and cotype312

18.7 Gaussian type and cotype314

18.8 Type and cotype of Lp spaces316

18.9 The little Grothendieck theorem revisited318

18.10 More on cotype320

18.11 Notes and remarks323