Introduction1 Measure and integral 1.1 Measure 1.2 Measurable functions 1.3 Integration 1.4 Notes and remarks2 The Cauchy-Schwarz inequality 2.1 Cauchy's inequality 2.2 Inner-product spaces 2.3 The Cauchy-Schwarz inequality 2.4 Notes and remarks3 The AM-GM inequality 3.1 The AM-GM inequality 3.2 Applications 3.3 Notes and remarks4 Convexity and Jensen's inequality 4.1 Convex sets and convex functions 4.2 Convex functions on an interval 4.3 Directional derivatives and sublinear functionals 4.4 The Hahn-Banach theorem 4.5 Normed spaces, Banach spaces and Hilbert space 4.6 The Hahn-Banach theorem for normed spaces 4.7 Barycentres and weak integrals 4.8 Notes and remarks5 The Lp spaces 5.1 Lp spaces, and Minkowski's inequality 5.2 The Lebesgue decomposition theorem 5.3 The reverse Minkowski inequality 5.4 HSlder's inequality 5.5 The inequalities of Liapounov and Littlewood 5.6 Duality 5.7 The Loomis-Whitney inequali'ty 5.8 A Sobolev inequality 5.9 Schur's theorem and Schur's test 5.10 Hilbert's absolute inequality 5.11 Notes and remarks6 Banach function spaces 6.1 Banach function spaces 6.2 Function space duality 6.3 Orlicz space 6.4 Notes and remarks7 Rearrangements 7.1 Decreasing rearrangements 7.2 Rearrangement-invariant Banach function spaces 7.3 Muirhead's maximal function 7.4 Majorization 7.5 Calder6n's interpolation theorem and its converse 7.6 Symmetric Banach sequence spaces 7.7 The method of transference 7.8 Finite doubly stochastic matrices 7.9 Schur convexity 7.10 Notes and remarks Maximal inequalities 8.1 The Hardy-Riesz inequality 8.2 The Hardy-Riesz inequality 8.3 Related inequalities 8.4 Strong type and weak type 8.5 Riesz weak type 8.6 Hardy, Littlewood, and a batsman's averages 8.7 Riesz's sunrise lemma 8.8 Differentiation almost everywhere 8.9 Maximal operators in higher dimensions 8.10 The Lebesgue density theorem 8.11 Convolution kernels 8.12 Hedberg's inequality 8.13 Martingales 8.14 Doob's inequality 8.15 The martingale convergence theorem 8.16 Notes and remarks9 Complex interpolation 9.1 Hadamard's three lines inequality 9.2 Compatible couples and intermediate spaces 9.3 The Riesz-Thorin interpolation theorem 9.4 Young's inequality 9.5 The Hausdorff-Young inequality 9.6 Fourier type 9.7 The generalized Clarkson inequalities 9.8 Uniform convexity 9.9 Notes and remarks10 Real interpolation 10.1 The Marcinkiewicz interpolation theorem: I 10.2 Lorentz spaces 10.3 Hardy's inequality 10.4 The scale of Lorentz spaces 10.5 The Marcinkiewicz interpolation theorem: II 10.6 Notes and remarks11 The Hilbert transform, and Hilbert's inequalities 11.1 The conjugate Poisson kernel 11.2 The Hilbert transform on 11.3 The Hilbert transform on 11.4 Hilbert's inequality for sequences 11.5 The Hilbert transform on T 11.6 Multipliers 11.7 Singular integral operators 11.8 Singular integral operators on 11.9 Notes and remarks12 Khintchine's inequality 12.1 The contraction principle 12.2 The reflection principle, and Lavy's inequalities 12.3 Khintchine's inequality 12.4 The law of the iterated logarithm 12.5 Strongly embedded subspaces 12.6 Stable random variables 12.7 Sub-Gaussian random variables 12.8 Kahane's theorem and Kahane's inequality 12.9 Notes and remarks13 Hypercontractive and logarithmic Sobolev inequalities 13.1 Bonami's inequality 13.2 Kahane's inequality revisited 13.3 The theorem of Lataa and Oleszkiewicz 13.4 The logarithmic Sobolev inequality on Dd 13.5 Gaussian measure and the Hermite polynomials 13.6 The central limit theorem 13.7 The Gaussian hypercontractive inequality 13.8 Correlated Gaussian random variables 13.9 The Gaussian logarithmic Sobolev inequality 13.10 The logarithmic Sobolev inequality in higher dimensions 13.11 Beckner's inequality 13.12 The Babenko-Beckner inequality 13.13 Notes and remarks14 Hadamard's inequality 14.1 Hadamard's inequality 14.2 Hadamard numbers 14.3 Error-correcting codes 14.4 Note and remark15 Hilbert space operator inequalities 15.1 Jordan normal form 15.2 Riesz operators 15.3 Related operators 15.4 Compact operators 15.5 Positive compact operators 15.6 Compact operators between Hilbert spaces 15.7 Singular numbers, and the Rayleigh-Ritz minimax formula 15.8 Weyl's inequality and Horn's inequality 15.9 Ky Fan's inequality 15.10 Operator ideals 15.11 The Hilbert-Schmidt class 15.12 The trace class 15.13 Lidskii's trace formula 15.14 Operator ideal duality 15.15 Notes and remarks16 Summing operators 16.1 Unconditional convergence 16.2 Absolutely summing operators 16.3 (p, q)-summing operators 16.4 Examples of p-summing operators 16.5 (p, 2)-summing operators between Hilbert spaces 16.6 Positive operators on 16.7 Mercer's theorem 16.8 p-summing operators between Hilbert spaces 16.9 Pietsch's domination theorem 16.10 Pietsch's factorization theorem 16.11 p-summing operators between Hilbert spaces 16.12 The Dvoretzky-Rogers theorem 16.13 Operators that factor through a Hilbert space 16.14 Notes and remarks17 Approximation numbers and eigenvalues 17.1 The approximation, Gelfand and Weyl numbers 17.2 Subadditive and submultiplicative properties 17.3 Pietsch's inequality 17.4 Eigenvalues of p-summing and (p, 2)-summing endomorphisms 17.5 Notes and remarks18 Grothendieck's inequality, type and cotype 18.1 Littlewood's 4/3 inequality 18.2 Grothendieck's inequality 18.3 Grothendieck's theorem 18.4 Another proof, using Paley's inequality 18.5 The little Grothendieck theorem 18.6 Type and cotype 18.7 Gaussian type and cotype 18.8 Type and cotype of LP spaces 18.9 The little Grothendieck theorem revisited 18.10 More on cotype 18.11 Notes and remarksReferencesIndex of inequalitiesIndex
· · · · · · (收起)