Review of Handbook of Analysis and its Foundations, by Eric Schechter

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REVIEW OF
Handbook of Analysis and its Foundations
by Eric Schechter
Academic Press, 1996; xxii + 883 pp.

Reviewer

J.R. Isbell

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A one-volume ``Handbook of Analysis'' is plainly not intended to update Dunford & Schwartz [DS]. So what is intended? The author of this book says (I adopt his nickname HAF for it) ``HAF is a self-study guide, intended for advanced undergraduates or beginning graduate students in mathematics.'' [DS] is something else entirely; and yet, there is enough likeness for a comparison to be of interest. The last two chapters of Volume I [DS] already go beyond HAF (mostly; HAF does have two pages on the spectrum of an operator, and nine pages on semigroups and dissipative operators). But the first twenty chapters of HAF correspond mostly to nothing in [DS]; some of this is prerequisite material which Dunford and Schwartz assumed known, but much of it is an introduction to a wider world. HAF is not, for the first seventeen chapters, a handbook.

Schechter says that an earlier version of this book was called Choice, Completeness, Compactness. The way this suggests of looking at existence theorems illuminates the unusual choice of contents for an analysis book. The student who has done some serious self-study with HAF will be notably knowledgeable about formal logic (always assuming that she retains a primary interest in analysis, unlike the converts Paul Cohen and Errett Bishop).

How dogged need the student be, to avoid dangerous skipping before she gets to the interesting bits? Not very. The book is fairly lively. On the first page of text, under the heading ``Mathematical Language and Informal Logic,'' we get ``All letters are variables, but some letters are more variable than others (as George Orwell might have put it).'' This sort of effervescence is frequent. It spills over, onto HAF's home page at http://math.vanderbilt.edu/~schectex/ccc/, where besides a few (identified) mathematical errors and typographical errors there is a tsunami of Addenda, or ``things that I now wish I could add to my book.''

The book is divided into quarters called ``Sets and Orderings,'' ``Algebra,'' ``Topology and Uniformity,'' ``Topological Vector Spaces.'' The principal novelties in the first part are (1) a very abstract Sperner Lemma after H. van Maaren, in [vM], (2) an unusually full statement of various (weak) forms of choice and ideas of constructivism, and (3) an elaborate hierarchy of (four) notions of subnet. The van Maaren lemma leads, 700 pages later, to Brouwer and Schauder fixed point theorems and beyond, e.g. to the Browder-Göhde-Kirk fixed point theorem for nonexpansive self-maps of a closed bounded convex set in a uniformly convex Banach space. There is a consistent concern with actual constructions. For instance, the first applications of van Maaren's lemma are theorems to the effect that for an arbitrary self-map of a sort of cell C there is a small set S with f(S) coming near S. After the Schauder theorem there comes another such Approximate Fixed Point Theorem. As for the various choice principles, note UF1-UF28: propositions equivalent to (UF1), every proper filter is contained in an ultrafilter. There are also HB1-HB26, equivalents of the Hahn-Banach Theorem. The facts that HB is weaker than UF, which is weaker than AC, are beyond the scope of HAF, but references are given. Turning now to subnets: Schechter's favorite version of subnet is that of J. F. Aarnes and P. R. Andenaes [AA]. In effect, one gives up any idea of a connecting mapping, and calls \N' an AA-subnet of \N if \N' is cofinally in every subset of the codomain in which \N is cofinally. (Equivalently, the associated filters satisfy \F'\supseteq \F.)

In the second part (``Algebra''), the reader needs to look out for special terminology affecting more than subnets. Ordered group means an abelian group with a translation-invariant partial order; if the order is total, it is a chain-ordered group. The category theory here is concrete category theory, and the universal algebra is Kuroshian, so a little special terminology is thereby required. The particular objective that brings in those theories is nonstandard analysis, or (for this review) NSA. A preliminary ``junior version'' of NSA is developed and applied in Chapters 9 and 10; the main product is a notion of hyperreal line considerably weaker than the usual one. (We must not say ``standard'' here!)

The second part culminates in the longest (by 85%) chapter in the book: 14, ``Logic and Intangibles.'' An important passage on the second page of the chapter runs ``For a first reading, some may choose to skip ahead to the end of this chapter and just read the summary of consistency results and the explanation of intangibles; the rest of this chapter will not be needed elsewhere in the book.'' The consistency results are substantially what is now known about relative consistency of eight or so seemingly inequivalent alternatives to AC and simple combinations of them. The explanation of intangibles is trickier. First, the relevant logic is intensional, not extensional; what is intangible is not an object x but a description \phi(x). For intangibility, (\exists x)\phi(x) should be a theorem of ZFC but there should be no ``explicit example'' of such an x. ``The only kind of intangible considered in most of this book'' is one whose existence is known not to be a theorem of ZF+DC. The reader who did not skip ahead will have had a sixty-page crash course in language, models, syntax/semantics, Boolean-valued models, and consistency and intangibility. Toward the end, the usual NSA is presented, basically as in [SL] -- though far, far more briefly. There is also a sketch of E. Nelson's alternative formulation of the same theory.

{It is unfortunate that there are not fuller references for Chapter 14. Much of the exposition there is totally inadequate for self-study. The references on consistency seem ample; on NSA or intangibles, derisory.}

The spirit of the seven-chapter part called ``Topology and Uniformity'' [whose last chapter is measure theory] tells us on page 455 that ``The preceding definitions of compactness and their proof of equivalence did not require the Axiom of Choice'' or any part of it; but using Choice, one can ... (and so on). If a reader who has not had a course in topology takes up this book, she may well be put off by the elaborate distinctions; they seem (at least, to a topologist) more distracting, less instructive, here than in other quarters. Indeed, the author has marked some passages ``Optional;'' not enough to remove the danger indicated, but it could encourage a diffident reader to take more options.

{A student --- aged seventeen or seventy --- who wants to sample this book to see if it is worth a serious reading might well take Chapter 17, ``Compactness,'' for the sample. It is concentrated Schechter, neither from the analytic center nor from the semi-constructivist periphery ofthe book.}

Chapter 18, ``Uniform Spaces,'' is where the book starts to feel like a traditional handbook. Naturally the main scene is metric spaces, and there we find besides the uniformly continuous maps and the nonexpansive maps the Lipschitz (-ian) and the Hölder continuous maps. The main attraction in Chapter 19, ``Metric and Uniform Completeness,'' is twelve pages of commentary about the Contraction Mapping Theorem: generalizations, Meyers' Converse describing self-mappings of a complete metric space X which become strict contractions by a suitable remetrization of X, and Bessaga's Converse which does the same thing on a bare (not Baire!) set X. Baire Theory is the next chapter. Indeed, the Baire and measure/integration chapters concluding the third part are handbooky in the best sense; somewhat Landauisch, with main lines clearly marked, crucial examples given, less essential material sketched, or covered by references. A sample of how Schechter, even in the shortest chapter since Chapter 2 (sc. Baire Theory), gives more than Landau: between Corollary 20.11 and its proof he Remarks, ``[the] condition is satisfied by bounded metric spaces that are not too irregularly shaped. For instance,'' a broad simple class of instances.

The first two chapters of the fourth part (22, ``Norms;'' 23, ``Normed Operators'') are the most classical in the book. Chapter 24, ``Generalized Riemann Integrals,'' points particularly to the last chapter (30, ``Initial Value Problems;'' a specialty of Schechter's). The generalized Riemann or Henstock integral is much as in Henstock [H], but confined to paths f:[a,b]-->X in Banach spaces. (The integral may be a Henstock-Stieltjes integral of f d\phi.) There are also Bochner integrals of vector-valued (Banach space-valued) functions on a measure space, briefly introduced in Chapter 23 and treated more fully in Chapter 29.

Chapter 25, ``Fréchet Derivatives,'' studies some basic ``physiology'' of derivatives and integrals: chain rule, (continuous) partial derivatives vs. Fréchet derivative, complex derivative and Cauchy-Riemann equations, Inverse Function Theorem and Implicit Function Theorem. These last two require continuous derivatives or, considerably more generally, strong derivatives at a point after Behrens [B] and Nijenhuis [N]. A bounded linear map L:X-->Y is the strong derivative f'(\xi) of f:\Omega-->Y, where \Omega\subset X, if

           ||f(x)-f(u)-L(x-u)||
   lim     -------------------- = 0.
x,u-->\xi       ||x-u||

Chapter 26, ``Metrization of Groups and Vector Spaces,'' takes its subject up to Pontryagin Duality and Haar Measure (Optional; Proofs Omitted) and further, to a very brief presentation of Banach lattices. Chapter 27, ``Barrels and Other Features of TVS's,'' is more briefly bornology. The fixed point theorems come here, and then the Barrel Theorems (barrel and ultrabarrel versions) concerning (mainly) Closed Graph and Uniform Boundedness Properties; also inductive topologies, needed notably for L. Schwartz' distributions. This chapter has a remarkable three-page concluding section called ``The Dream Universe of Garnir and Wright.'' It presents results of H. G. Garnir [G] and J. D. M. Wright [W] such as ``Every linear operator from a Fréchet space to a topological vector space is continuous'' and (therefore) ``Any two complete norms on a vector space are [topologically, hence] Lipschitz equivalent,'' which are not ``true'' as we usually use the word, i.e. not theorems of ZFC. They are, however, irrefutable, being implied by ZF+dependent choice+``Every subset of R has the Baire property,'' which S. Shelah has shown [S] to be consistent if ZF is consistent.

Chapter 28, ``Duality and Weak Compactness,'' begins with equivalents 17 through 23 of the Hahn-Banach Theorem, mostly in terms of separation in locally convex spaces X. The pinnacle of the chapter is R. C. James' ``Sup Theorem,'' the point of which is that a bounded weakly closed set B in X is weakly compact if every functional in X* attains a maximum on B [J]. This wraps up in one package several considerable theorems on weak compactness. Prominent in Chapter 29, ``Vector Measures,'' are the Radon-Nikodym Property (the classical Radon-Nikodym Theorem is that a 1-dimensional space has the Property) and more results about weak forms of choice and the anti-choice principle ``Every subset of R has the Baire property.'' The final chapter, 30 ``Initial Value Problems,'' is the shortest since Chapter 23; a reader who, like Oliver Twist, wants `More,' is referred to [Sch].

This book will be a valuable resource for the ambitious students at whom it is aimed and for a number of licensed mathematicians, outside analysis and perhaps inside, who are interested in broadening their perspectives. We are all in the author's debt.

Bibliography

[AA] J. F. Aarnes and P. R. Andenaes, ``On nets and filters,'' Math. Scand. 31, 1972, 285-292.

[B] M. Behrens, ``A local inverse function theorem,'' in Victoria Symposium on Non-standard Analysis (University of Victoria 1972), Lecture Notes in Math. 369, Springer-Verlag, Berlin, 1974, 34-36.

[DS] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Pure Appl. Math. 7, Wiley, New York, 1957.

[G] H. G. Garnir, ``Solovay's axiom and functional analysis,'' in Functional analysis and its applications (International Conference, Madras, 1973), Lecture Notes in Math. 399, Springer-Verlag, Berlin, 1974, 189-204.

[H] R. Henstock, The General Theory of Integration, Clarendon Press, Oxford, 1991.

[J] R. C. James, ``Weakly compact sets,'' Trans. Amer. Math. Soc. 113 (1964), 129-140.

[vM] H. van Maaren, ``Generalized pivoting and coalitions,'' in The Computation and Modelling of Economic Equilibria, Contrib. to Econom. Anal. 167, North-Holland, 1987, 155-176.

[N] A. Nijenhuis, ``Strong derivatives and inverse mappings,'' Amer. Math. Monthly 81 (1974), 969-980; addendum, ibid. 83 (1976), 22.

[Sch] E. Schechter, ``A survey of local existence theories for abstract nonlinear initial values problems,'' in Nonlinear semigroups, partial differential equations and attractors (Washington, D. C. 1987), Lecture Notes in Math. 1394, Springer-Verlag, Berlin, 1989, 136-184.

[S] S. Shelah, ``Can you take Solovay's inaccessible away?,'' Israel J. Math. 48 (1984), 1-47.

[SL] K. Stroyan and W. Luxemburg, Introduction to the Theory of Infinitesimals, Academic Press, 1976.

[W] J. D. M. Wright, ``Functional analysis for the practical man,'' in Functional Analysis: Surveys and Recent Results (Conference on Functional Analysis, Paderborn, 1976, North-Holland Math. Stud. 27, Amsterdam, 1977, 283-290.

This document was last revised on July 25, 1997.