Half-title -- Title -- Copyright -- Contents -- Introduction -- 1 Measure and integral -- 1.1 Measure -- 1.2 Measurable functions -- 1.3 Integration -- 1.4 Notes and remarks -- 2 The Cauchy--Schwarz inequality -- 2.1 Cauchy's inequality -- 2.2 Inner-product spaces -- 2.3 The Cauchy--Schwarz inequality -- 2.4 Notes and remarks -- Exercises -- 3 The arithmetic mean-geometric mean inequality -- 3.1 The arithmetic mean-geometric mean inequality -- 3.2 Applications -- 3.3 Notes and remarks -- Exercises -- 4 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 remarks -- Exercises -- 5 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 Hölder's inequality -- 5.5 The inequalities of Liapounov and Littlewood -- 5.6 Duality -- 5.7 The Loomis--Whitney inequality -- 5.8 A Sobolev inequality -- 5.9 Schur's theorem and Schur's test -- 5.10 Hilbert's absolute inequality -- 5.11 Notes and remarks -- Exercises -- 6 Banach function spaces -- 6.1 Banach function spaces -- 6.2 Function space duality -- 6.3 Orlicz spaces -- 6.4 Notes and remarks -- Exercises -- 7 Rearrangements -- 7.1 Decreasing rearrangements -- 7.2 Rearrangement-invariant Banach function spaces -- 7.3 Muirhead's maximal function -- 7.4 Majorization -- 7.5 Calderón'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 -- Exercises.

8 Maximal inequalities -- 8.1 The Hardy--Riesz inequality… -- 8.2 The Hardy--Riesz inequality (p=1) -- 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 remarks -- Exercises -- 9 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 remarks -- Exercises -- 10 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 remarks -- Exercises -- 11 The Hilbert transform, and Hilbert's inequalities -- 11.1 The conjugate Poisson kernel -- 11.2 The Hilbert transform on L2(R) -- 11.3 The Hilbert transform on Lp(R) for… -- 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 Lp(Rd) for… -- 11.9 Notes and remarks -- Exercises -- 12 Khintchine's inequality -- 12.1 The contraction principle -- 12.2 The reflection principle, and Lévy'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 remarks -- Exercises -- 13 Hypercontractive and logarithmic Sobolev inequalities -- 13.1 Bonami's inequality -- 13.2 Kahane's inequality revisited -- 13.3 The theorem of Latala and Oleszkiewicz -- 13.4 The logarithmic Sobolev inequality on… -- 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 remarks -- Exercises -- 14 Hadamard's inequality -- 14.1 Hadamard's inequality -- 14.2 Hadamard numbers -- 14.3 Error-correcting codes -- 14.4 Note and remark -- 15 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 remarks -- Exercises -- 16 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 L1 -- 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 remarks -- Exercises -- 17 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 remarks -- Exercises -- 18 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 remarks -- Exercises -- References -- Index of inequalities -- Index.