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Methods in classical and functional analysis. (English) Zbl 0223.46001
Addison-Wesley Series in Mathematics. Reading, Mass. etc.: Addison-Wesley Publishing Company. ix, 486 p. £6.55 (1972).
Summary: What the author wanted to do and achieved in a very striking way is said in the preface from which I quote: “Modes come and go in mathematics as in most fields. During the half-century and more that I have worked in the vineyard I have heard many dire predictions for the fate of my ideas and interests. Abstraction has been in the saddle during most of the time and has ridden us mercilessly. In a modest way I have taken part in this development. I did not believe in abstraction per se, one should know what one is trying to generalize and one should show that the generalization is significant. I have tried to keep at least one foot an the ground while craning my neck to look into Heaven? What is Heaven? There are some doubts, and the more extravagant claims of the abstract mathematicians to be the sole dispenser of the true faith and the arbiters of values are received with a healthy scepticism. $$\ldots$$ This book may be regarded as part of the backlash. If the book has a thesis, it is that a functional analyst is an analyst, first and foremost, and not a degenerate species of a topologist. His problems come from analysis and his results should throw light on analysis. $$\ldots$$ It seemed to me that I could do some useful work in giving the student a historical perspective and in showing how the multitude of abstract concepts have arisen and are present in nuce in Euclidean spaces.”
The reviewer considers the book elementary in the sense that there are virtually no prerequisites required but it leads by simple generalizations to advanced mathematics. Not the most generality in theorems is sought but the generality of methods is presented. There are lots of historical remarks giving names and duration of life span of mathematicians contributing to the specific subject. Each chapter has its exercises and a column “collateral reading” quoting textbooks and classical research papers. The topics are in the order of the book: Complex Euclidean spaces, linear algebra and an extensive treatment of the resolvent concept for matrices so preparing the soil for the corresponding operator concept. Banach and Hilbert spaces, linear transformations and functionals, spaces of linear operators, sequence spaces, function spaces. Starting with lack of completeness with respect to the Riemann integral, Lebesgue measure, integration and Lebesgue spaces $$L^p$$ are introduced. Application to Fourier series are discussed. Metric spaces, contraction mappings and fixed point theorems are applied to integral equations. E. Landau’s inequality (1913) for real functions $$| f'|^2\leq 4\,| f|\, | f''|$$ appears in connection with R. R. Kallman and G. C. Rota’s (196?) result $$\| Ax\|^2\leq 4\,\| x\|\, \| A^2x\|$$, where $$A$$ is the infinitesimal generator of any contraction semigroup. Existence and uniqueness concepts from implicit function theorem to ordinary differential equations are presented.
Under the headline real and complex analysis in linear spaces you find: the principle of uniform boundedness, the different topologies, vector and operator valued functions, abstract holomorphic functions, Bochner integrals, Fréchet differentiability etc.
The next chapters deal with Banach algebras – Gel’fand’s theorem, operational calculus, linear transformation – closure, adjoints, spectra resolvent, inner product space – numerical range, spectral theorem, operational calculus. Although these themes have been dealt with in the first chapters they are taken up again, looked at in a more general fashion, and new applications come out.
Two chapters are devoted to functional inequalities containing famous results of C. Carathéodory, M. Nagumo and Gronwall’s Lemma. Application to differential equations are given. Subadditive functions, convex functions and non-archimedean valuations occur as examples for functional inequalities on product spaces.
A chapter about functional equations starts with the technique of getting a priori information out of special equations. This technique is called “cryptoanalysis”, another example for the picturesque language of the author. Cauchy’s equation are dealt with and Hamel basis is introduced where it first showed up. Finally, there is some information about mean value operators again starting with trivial examples as arithmetic and geometric means and leading to potential theories. All in all a fancy bock.

##### MSC:
 46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis 46A03 General theory of locally convex spaces 46B03 Isomorphic theory (including renorming) of Banach spaces 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 46Exx Linear function spaces and their duals 46Gxx Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces)