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Inequalities: A Journey into Linear Analysis

D. J. H. Garling
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

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, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.


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 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 L2(R)
11.3 The Hilbert transform on Lp(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 Lp(Rd) 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 D2d
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 Lp spaces
18.9 The little Grothendieck theorem revisited
18.10 More on cotype
18.11 Notes and remarks


Index of inequalities