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
3  The AM-GM inequality
  3.1 The AM-GM inequality
  3.2  Applications
  3.3  Notes and remarks
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
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  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 remarks
6  Banach function spaces
  6.1  Banach function spaces
  6.2  Function space duality
  6.3  Orlicz space
  6.4  Notes and remarks
7  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 remarks
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
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
11  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 remarks
12  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 remarks
13  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 remarks
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
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
  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
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
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
References
Index of inequalities
Index