to

\begin{matrix}K̃_{s,p} = H̃^{s−2m,p,(r−2m)}(G) \\ × ∑_{j=1}^{m} B^{s−m_{j} −1/p,p}(∂G), \end{matrix}

where H^{t,p}(G) and B^{t,p}(∂G) are Sobolov and Besov spaces, respectively. The theorem states that for each s, p described, A is a Fredholm operator (i.e., A has a finite dimensional null space, R(A) is closed and has finite codimension). Moreover, the null spaces of A and A^{∗} are independent of s and p. This theorem contains most, if not all, of the results for this problem obtained during this period. The author proceeds to give a full proof of the theorem and its consequences. The book is essentially selfcontained. After a general introduction (Chapter 0), the author describes various function spaces (Chapter 1), the specific spaces used (Chapter 2), and elliptic boundary value problems (Chapter 3). He then states and proves his main theorem (Chapter 4). The remainder of the book considers special topics. This includes problems with normal boundary conditions (Chapter 5), traces on the boundary (Chapter 6), local increases in smoothness (Chapter 7), degenerate elliptic problems (Chapter 8), problems depending on parameters (Chapter 9), and systems (Chapter 10). Bibliographical notes are given at the end.

The book is well written and deals well with a topic which requires great care and attention to detail. It is an excellent digest of known results for this problem and a valuable reference for those who wish to apply known results for problem (1) to various other disciplines.

MARTIN SCHECHTER

University of California, Irvine

Handbook of Analysis and Its Foundations. By Eric Schechter. Academic Press, New York, 1997. xxii+883 pp. $74.95, hardback. ISBN 0-12-622760-8.

Every once in a while a book comes along that so effectively redefines an educational enterprise—in this case, graduate mathematical training—and so effectively reexamines the hegemony of ideas prevailing in a discipline—in this case, mathematical analysis—that it deserves our careful attention. This is such a book. There is nothing else remotely similar to it in any of the current books on integration, real analysis, set theory, or any other related subject. It is colossal, invigorating, and refreshingly contemporary in its concerns. Many of the items in the bibliography of the book were published after 1990, and many, many after 1980. Most of us will be unfamiliar with a substantial number of the theorems, techniques, and tools the author utilizes. The material relating to philosophical issues will be familiar to some of us from our reading in general interest mathematical publications, for instance, The Mathematical Intelligencer—the main ideas of constructivism, nonstandard analysis, or how mathematical analysis changes if its settheoretic underpinnings are altered. But nowhere are these matters treated coherently in a book that is intended for use as a handbook for pure or applied mathematicians. All in all, this book reaffirms for us how subtle, beautiful, and joyous mathematics can be, although, as the author points out through a sly quotation from the novelist Gabriel Garcia Márquez, one of many epigraphs, subtlety itself can be a delusive goal:

He liked to wander through metaphysical obstacle courses. That was what he was doing when he used to sit in the bedroom. … However, he himself had become so subtle in his thinking that for at least three years in his meditative moments he was no longer thinking about anything.

Why should SIAM Review readers care about this book? Well, a lot of the material will be of interest to all mathematicians— pure as well as applied, and how many among us have escaped the traditional real variables sequence? Also the author’s many observations about the pursuit of applied mathematics are provocative and sometimes unsettling. He has attempted, successfully, I think, to avert the reader’s succumbing to the technical doldrums, emphasizing theorems that have short, elegant, or highly intuitive proofs. He has included only a few lengthy proofs, that of Brouwer’s theorem, for instance.

The book is divided into four parts, each containing a number of chapters. Let me itemize the chapter titles, although to do so gives little hint of the many surprises the author has in store for us:

A. SETS AND ORDERINGS: sets; functions; relations and orderings; more about sups and infs; filters, topologies, and other sets of sets; constructivism and choice; nets and convergences.

B. ALGEBRA: elementary algebraic systems; concrete categories; the real numbers; linearity; convexity; Boolean algebras; logic and intangibles.

C. TOPOLOGY AND UNIFORMITY: topological spaces; separation and regularity axioms; compactness; uniform spaces; metric and uniform completeness; Baire theory; positive measure and integration.

D. TOPOLOGICAL VECTOR SPACES: norms; normed operators; generalized Riemann integrals; Fréchet derivatives; metrization of groups and vector spaces; barrels and other features of TVSs; duality and weak compactness; vector measures; initial value problems.

The reader will find among the above many of the topics that are covered in the usual graduate course in real analysis, but there are quite a few unexpected entries. A lot of topology and functional analysis, that’s understandable, perhaps. But constructivism? Barrels? Initial value problems? Category theory (of the MacLane–Eilenberg sort)? The attention to foundational problems is surprising, too, and the development of integration employs a tool unfamiliar to many of us, the Henstock integral. Other unusual features are a thorough discussion of filters, symmetric and preregular spaces, nonstandard analysis, converses to Banach’s contraction theorem, formal logic, model theory. Pity the beleagured applied mathematician, continually hounded to learn more and more about pure mathematics to effectively pursue his temporal concerns; he or she may see this book as very minatory, since it seems to imply that barrel spaces are necessary to understand fluid dynamics, elasticity, or other things of practical value. Well, perhaps not barrel spaces, but as our concern with nonlinear fluid low and with ill-posed problems escalates, weak convergence, weak topologies, and generalized solutions are becoming of increasing practical significance. The author deals with all of these.

The author quotes Halmos, who said that one good way to learn a lot of mathematics is by reading the first chapters of many books, and then asserts

I have tried to improve upon that collection of first chapters by eliminating the overlap between separate books, adhering to consistent notation, and inserting frequent cross referencing between the different topics.

Obviously and regrettably, I don’t have the space to go into detail about each and every topic in the book; instead, I will skirt from mountain top to mountain top, picking out those peaks which I think especially contribute to the book’s extraordinary presence.

Because existence proofs in all areas of mathematics generally use either compactness, completeness, or the axiom of choice, the author gives those subjects special attention. In the first chapter he establishes the standard basic set theory and then defines Zermelo–Fraenkel set theory (ZF) and the axiom of choice (AC). Other axioms that are sometimes appended to ZF are BP (every subset A of the reals can be written A = GΔM, where G is open and M is of first category), CH (the continuum hypothesis), LM (every subset of the reals is Lebesgue measurable), DC (the axiom of dependent choice; the definition is a little technical, but it means, roughly, that the choices of set elements depend on previous choices). Later, in Chapter 14, the author utilizes these concepts to state recent dramatic results concerning the consistency of various versions of set theory. For instance a famous result of Cohen (1963) is that if (ZF) is consistent, then so is (ZF + not AC + not CH). If (ZF) is consistent, so is (ZF + BP + LM) (Solvay, 1964/1965). That (ZF + AC) implies (not BP) is established in this book. Throughout the book, the author assumes the consistency of conventional set theory (ZF + AC) and the consistency of Shelah’s alternative, (ZF + DC + BP). The material on foundations is deep, and may cause cluster headaches for the reader not used to thinking in the patois logicians use. The author’s discussion, though, is casual and friendly (something characteristic of his style throughout) and meticulously motivated.

In Chapter 6 the author discusses constructivism. A mathematician is behaving as a constructivist if he or she insists that any legitimate mathematical proof must demonstrate a way to find the object in question. Constructivists, thus, repudiate the axiom of choice and the axiom of regularity.^{1} Of course, applied mathematicians are arrant constructivists, but the author points out that any mathematician acting in a pedagogical capacity is a closet constructivist. The discussion, elegant and pointed, is the most satisfying reflection on constructivism I have ever seen:

Most mathematicians are parttime informal constructivists in this respect: In teaching or learning mathematics, we try to follow any abstract idea with one or more concrete examples. … We follow that pedagogical practice—i.e., of giving examples—whenever possible, but we must depart from that practice at times, for some mathematical ideas are inherently nonconstructive. Indeed, some of the objects studied in this book are intangible:^{2} We shall see that the objects “exist,” but that ex-

1Too technical to define here. It is important only for systems in which there is possible a very deep nesting of sets, and so has little effect on ordinary mathematics.

2An “intangible” is an object whose existence can be proved in conventional mathematics (ZF + AC) but not in the weaker quasiconstructive mathematics (ZF + DC).

3I am thinking of the incident in which a revolutionary new calculus book that featured nonstandard analysis was delivered to a rabid constructivist to review for Mathematical Reviews. I thought the book was adventurous and exciting; the reviewer, however, succumbed to territoriality and gave the book a thrashing.

plicitly constructible examples of these objects do not exist.

He goes on to point out that often nonconstructive arguments can be replaced by constructive ones, but only occasionally with any ease. Consider the following proposition:

There exist positive irrational numbers a and b such that a^{b} is rational. The easiest proof is nonconstructive.

Either

(I) \sqrt{2}^{\sqrt{2}} is rational—then take a = b = \sqrt{2}, or

(II) \sqrt{2}^{\sqrt{2}} is irrational—then take a =

\sqrt{2}^{\sqrt{2}} and b = \sqrt{2}.

Most of us will feel cheated by this proof. Clamoring to know which option, (I) or (II), prevails, we lose interest in the question being decided. Well, it’s the second: \sqrt{2}^{\sqrt{2}} is irrational. However, the (constructive) proof is much much longer. (The Gelfond– Schneider theorem states that if a and b are algebraic numbers, a ≠ 1, b irrational, then a^{b} is transcendental.)

Section 6.8 is entitled Constructivism Versus Mainstream Mathematics. It is the most insightful and fair-minded appraisal I have yet seen of the differences between the constructivist and the nonconstructivist world-views, whose proponents have sometimes defended their ideas with belligerence and occasional deviousness.^{3} 99.99 % of mathematicians of the theoretical variety are, in the pursuit of their trade, nonconstructivists. However, Schechter points out that the badmouthing most of us give the constructivists may be less germane to the conflict than is the different language the camps use.

Despite its growing literature, constructivism remains separated from the mainstream of mathematics. This may be largely because constructivism’s finer distinctions necessitate a use of language quite different from, and more complicated than, that of the mainstream mathematician. For instance, among some constructive analysts, x ≠ y means simply the negation of x = y while x # y means the slightly stronger condition of apartness: we can find a positive lower bound for the distance between approximations to x and y. Thus constructivists distinguish between notions that the classical mathematician is accustomed to viewing as identical. Consequently a mainstream mathematician can only learn constructivism by relearning his or her entire language—a sizeable undertaking.

Part of the author’s plan is to vary the quantity of constructiveness employed depending on the object being studied, but he admits that weakening AC can entail the paradoxical:

In particular, we may compare results using the Axiom of Choice with results that only require a weakened form of the Axiom of Choice. At first glance, that looks like a rather strange notion; after all, either we can find a certain mathematical object, or we can’t. How can we say that one object is harder to find than another object, when, in fact, we can’t find either of them?

Essentially, the author claims the right to accept AC on Tuesdays but only a modified version on Wednesdays. One only has to mull the above passages over a bit to realize how daring and innovative is the author’s approach to his material. At no point in this very large book is the reader allowed to forget the set-theoretic basis, more precisely, the logico-philosophical basis, upon which all mathematical analysis ultimately depends. One might term this the vertical approach, versus the horizontal approach, to mathematical exposition, fugue versus monody. Foundational issues inform all the developments in this book.

Section 7 deals with various ideas of convergence, starting with ordinary convergence (in a metric space). The author observes that such convergence lacks, for many applications, sufficient generality, then leads us step by step through nets, universal nets, convergence in posets, convergence in complete lattices. Because of the leisurely verbal pace, even those new to this material will respond to its inherent beauty.

Long after the discussion of Lebesgue theory, the author deals with generalized integration in a chapter called Generalized Riemann Integrals, but which actually covers a number of hypermodern developments in integration theory. The foundation of the chapter is the Henstock integral (introduced independently at about the same time in the 1950s by Henstock and Kurzweil). For those unfamiliar with this astonishing instrument, I give the definition:

Let f : [a, b] → X where X is a normed vector space, and let v ∈ X. We say that v is a Henstock integral of f over [a, b] if for each number ε > 0 there exists some function δ : [a, b] → (0, +∞) such that if n ∈ N and a = t_{0} ≤ t_{1} ≤ t_{2} ≤ ⋯ ≤ t_{n} = b and τ_{j} ∈ [t_{j−1}, t_{j}] with t_{j} − t_{j−1} < δ(τ_{j}) for all j then

(1)

∥v − ∑_{j=1}^{n}(t_{j} − t_{j−1})f(τ_{j})∥ < ε.

This definition at first glance—and the author admits as much—may seem excessively (if for…there exists…such that if…with…for all…then) convoluted. But on rereading, you’ll find yourself accommodating to it.^{4} Let δ(t) ≡ δ = const. and you have the classical Riemann integral. It is surprising that an integral so transparently similar to the Riemann integral could be so much more general—there are functions that are Henstock integrable that are not Riemann integrable. Let

f(t) = \{ \begin{matrix}0, & t = 0, \\ t^{−1/2}, & t ≠ 0; \end{matrix}

then f is not (properly) Riemann integrable on [0, 1], but it is Henstock integrable. An amazing idea, putting that little (t) after the δ.

4If I were presenting this material to a class, I would have defined a dissection of [a, b] as a set of partition points {t_{j}} plus a set of intermediate points {τ_{j}}. Then v is a Henstock integral of f if for each number ε > 0 there exists a positive function δ for which (1) holds for all dissections satisfying t_{j} − t_{j−1} < δ(τ_{j}).

It can be shown the Henstock integral is unique and, like the Lebesgue integral, has monotone convergence properties. More surprisingly, the Henstock integral contains the Lebesgue integral. For instance, let

f(t) = \{ \begin{matrix}0, & t = 0, \\ t^{2} cos ( \frac{π}{t^{2}} ) , & t ≠ 0. \end{matrix}

The derivative f′(t) is Henstock, but not Lebesgue, integrable on [0, 1].

The author discusses a number of other modern integrals (the Denjoy–Perron integral, the Bochner–Lebesgue integral, etc.) and illustrates their relation to the Henstock device. The Bochner/Lebesgue approach is more powerful, insofar as it is applicable to a wider variety of measure spaces (Ω, δ, μ), but in other respects the Henstock integral is more general. The Bochner–Lebesgue integral implies a separability condition on the underlying Banach space and an absolute integrability condition; the Henstock integral assumes neither.

I thought the book had morphed into a sci-fi narrative when I encountered the section, “The Dream Universe of Garnir and Wright.” It’s all legitimate mathematics, however. The author wishes to examine how analysis is affected when one replaces conventional set theory (ZF + AC) with an alternative version of set theory; this helps to explain those intangibles—objects that exist but lack constructible examples; see the previous footnote—encountered in conventional set theory. H. G. Garnir used the term “dream space” to denote any normed space X with the property that every linear map from X into a normed space is continuous. Of course, all finite dimensional spaces are dream spaces. Under the hypotheses of conventional set theory, there are no other dream spaces. But what if the rules are changed? In particular, what if the Shelah axioms (ZF + DC + BP) are adopted? The effect is dramatic, to say the least.

The author has been preoccupied with the contrasting philosophies of pure and applied mathematicians; strangely, it is precisely in this bizarre and abstruse realm of dream spaces that one of the most fundamental differences between these worldviews stands illuminated.

Applied mathematicians do not use the Axiom of Choice, and so they cannot prove the existence of inequivalent complete norms on a vector space. More precisely, they cannot construct two complete norms and prove that those norms are inequivalent. … They may also be unable to prove that the two norms are equivalent. Thus, it is conceivable that an applied mathematician could equip a vector space with two different, complete, usual norms which are not known (by the tools of applied mathematics) to be equivalent or inequivalent.

Since norms measure how close two things are, or, more critically, how large the error of an approximation is, it is ominous that applied mathematicians working with different norms may have drastically different perceptions of the validity of their results. The last chapter, initial value problems, deals with differential equations in a Banach space,

(2)  \begin{matrix}u′(t) = f(t, u(t)), 0 ≤ t ≤ T, \\ u(0) = x_{0}, \end{matrix}

where f is a given mapping from R × X into X, a subject of immense practical importance. When f does not depend on t explicitly, the system is called autonomous, otherwise, nonautonomous. The problem is to find the function u satisfying (2), called the solution of the initial value problem (2). The concerns addressed in this chapter are quite current. Although our knowledge about solutions is still a little inchoate, the state of affairs is, roughly, as follows:

(1) When f is continuous and X is finitedimensional, a solution exists (but not necessarily a unique one).

(2) In infinite dimensional spaces, the continuity of f is not sufficient to guarantee a solution, but “most” differential equations have unique solutions. (The delineation of “most” entails some technicalities.)

(3) Under certain additional assumptions (Lipschitz conditions, compactness, isotonicity, dissipativeness) which need not include continuity of f, solutions will be guaranteed.

It may be greedy, however, to insist that the system have a solution in the classical sense. There are other ways to specify what one means by a solution, and the author devotes some time to one of the cleverest and most productive of these: a Carathéodory solution. Let X be a Banach space, f a function from some subset of R×X into X. By a Carathéodory solution of (2) on an interval J we mean a function u : J → X such that Graph (u) ⊆ Dom (f) and the Bochner integral

∫_{a}^{b} f(t, u(t))dt, a < b in J,

exists and equals u(b) − u(a). Very surprisingly, a Carathéodory solution is continuous and for almost all t ∈ J, u′(t) exists and equals f(t, u(t)). (Whenever I see mathematicians so resourcefully finessing austere reality this way, I’m reminded of the folktale of making soup from stones.)

The question of whether such a solution exists globally in time can be divided among two different issues: do solutions exist at least locally in time? and, can such solutions be continued further in time? Necessary and sufficient conditions for the solution to be unique, even locally, however, are not yet known. The rest of the chapter explores conditions—Lipschitz, compactness, and isotonicity—that can ameliorate the difficulties of establishing existence and uniqueness.

Let me talk about the prose style of the book. The author has deliberately departed from the traditional real-variable textbook rhetoric—the abbreviated definition– theorem–proof approach with a minimum of intercalated commentary. The author’s way will not be to everyone’s taste. He writes exceedingly well, but the pace is measured, zealously well motivated, armed with historical observations, and often delightfully colloquial. It is antipodal to the style, say, of Royden or Rudin. I haven’t talked to many mathematicians weaned on Rudin/Royden and the like who actually favored those books so much; rather, they felt learning from them was a necessary scholarly rite of passage, a little like a fraternity hazing. Judging from the books I’ve been seeing lately, I think the fashion for concision has passed. I, for one, welcome a return to the more effusive writing of the great British mathematical books from the early part of this century. The author’s style probably points the way to the future. By his own account, he has attempted to write a self-contained but advanced book for self-directed study, a “summer vacation” book. He has tried to create the book he wishes he had before he began his own graduate studies, one, undoubtedly, that provides a charitable, rather than an exigent, foundation for a life-long professional career.

Physically, the book is beautiful, with wide pages, impeccable design, striking binding, and gorgeous mathematical typesetting. Academic Press is spoiling us. I commend Academic for its gumption in publishing this very atypical work.

There are things I didn’t like about the book. I accept that the book is not a primer, but it needs many more examples. The illustrations of the Fréchet derivative are too trivial; I would have welcomed examples applicable to some of the knotty operator equations one runs into in practice. A nice application would have been the use of the Fréchet derivative in Newton-type iterative methods for finding the solution of equations in normed spaces. Also, there are far too few exercises. The index is occasionally inaccurate and too scant to be very useful; I found that if I wished to relocate material for this review, it was usually of little help. Perhaps these blemishes will be repaired in future editions, and the book deserves them; it should be continually before us.

JET WIMP

Drexel University

Elliptic Marching Methods and Domain Decomposition. By Patrick J. Roache. CRC Press, Inc., Boca Raton, FL, 1995. $69.95. 190 pp., hardcover. ISBN 0-8493-7378-6.

Numerical methods for elliptical equations are considered to be quite mature. It is common knowledge that very large problems should be solved iteratively. In today’s scientific computation, finite element multigrid methods combined with domain