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Publisher:

Cambridge University Press

Publication Date:

2007

Number of Pages:

335

Format:

Paperback

Price:

45.00

ISBN:

9780521699730

Category:

Monograph

[Reviewed by , on ]

Allen Stenger

07/15/2008

This is a very deep look at a myriad of inequalities that are useful in linear functional analysis. The stated prerequisites are mild, but in fact the book moves so far and goes so deep that unless you are already well grounded in real analysis and functional analysis, you will quickly become lost. It is a book for specialists and not for the more general mathematical audience it seems to be pitched at.

Similarly this is not a book for beginners in inequalities, even though it starts out with familiar inequalities such as the arithmetic mean-geometric mean inequality. If your interest in primarily in inequalities rather than functional analysis, I recommend Steele's The Cauchy-Schwarz Master Class.

The author claims that the book shows inequalities in context, but I thought this was mostly unsuccessful. This is a typical functional analysis book in that way: results are presented and proved and never seen again, so you don't get a good idea why the results are interesting. There are exceptions, and there are some interesting historical discussions of how these results came about, but in general you don't know where these results came from or where they are going.

On the plus side, the book does include an enormous amount of material and is clearly written. It has a good index and the sources of the theorems are well-documented. Unfortunately the index does not include symbols, and this is a book that really needs a symbol index because of the sheer quantity of special symbols.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

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 L^{p} spaces

5.1 L^{p} 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

6 Banach function spaces

6.1 Banach function spaces

6.2 Function space duality

6.3 Orlicz spaces

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 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

8 Maximal inequalities

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

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

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 L^{2}(**R**)

11.3 The Hilbert transform on L^{p}(**R**) for 1 < p < ∞

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 L^{p}(**R**^{d}) for 1 ≤ p < ∞

11.9 Notes and remarks

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

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 D_{2}^{d}

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 L1

16.7 Mercer's theorem

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

16.9 Pietsch's domination theorem

16.10 Pietsch's factorization theorem

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

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 L^{p} spaces

18.9 The little Grothendieck theorem revisited

18.10 More on cotype

18.11 Notes and remarks

References

Index of inequalities

Index

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