HANDBOOK OF ANALYSIS AND ITS FOUNDATIONS is a book for beginning math graduate students. It is eventually about real and functional analysis, but it begins with a few hundred pages of prerequisite material from set theory, logic, algebra, and topology. This long web page is the book's preface, reformatted slightly for the web. You can skip ahead to any of these sections: About the Choice of Topics - Existence, Examples, and Intangibles - Abstract versus Concrete - Order of Topics - How to Use This Book - Acknowledgments - To Contact Me

# PREFACE

## About the Choice of Topics

Handbook of Analysis and its Foundations -- hereafter abbreviated HAF -- is a self-study guide, intended for advanced undergraduates or beginning graduate students in mathematics. It will also be useful as a reference tool for more advanced mathematicians. HAF surveys analysis and related topics, with particular attention to existence proofs.

HAF progresses from elementary notions -- sets, functions, products of sets -- through intermediate topics -- uniform completions, Tychonov's Theorem -- all the way to a few advanced results -- the Eberlein- Smulian- Grothendieck Theorem, the Crandall- Liggett Theorem, and others. The book is self-contained and thus is well suited for self-directed study. It will help to compensate for the differences between students who, coming into a single graduate class from different undergraduate schools, have different backgrounds. I believe that the reading of part or all of this book would be a good project for the summer vacation before one begins graduate school in mathematics. At least, this is the book I wish I had had before I began my graduate studies.

HAF introduces and shows the connections between many topics that are customarily taught separately in greater depth:

set theory, metric spaces, abstract algebra, formal logic, general topology, real analysis, and linear and nonlinear functional analysis, plus a small amount of Baire category theory, Mac Lane- Eilenberg category theory, nonstandard analysis, and differential equations.

Included in these customary topics are the usual nonconstructive proofs of existence of pathological objects. Unlike most analysis books, however, HAF also includes some chapters on set theory and logic, to explain why many of those classical pathological objects are presented without examples.

HAF contains the most fundamental parts of an entire shelf of conventional textbooks. In his "automathography," Halmos [1985] said that one good way to learn a lot of mathematics is by reading the first chapters of many books. 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. HAF's integrated approach shows connections between topics and thus partially counteracts the fragmentation into specialized little bits that has become commonplace in mathematics in recent decades. HAF's integrated approach also supports the development of interdisciplinary topics, such as the "intangibles" discussed later in this preface.

The content is biased toward the interests of analysts. For instance, our treatment of algebra devotes much attention to convexity but little attention to finite or noncommutative groups; our treatment of general topology emphasizes distances and meager sets but omits manifolds and homology. HAF will not transform the reader into a researcher in algebra, topology, or logic, but it will provide analysts with useful tools from those fields.

HAF includes a few "hard analysis" results: Clarkson's Inequalities, the Kobayashi- Rasmussen Inequalities, maximal inequalities for martingales and for Lebesgue measure, etc. However, the book leans more toward "soft analysis" -- i.e., existence theorems and other qualitative results. Preference is given to theorems that have short or elegant or intuitive proofs and that mesh well with the main themes of the book. A few long proofs -- e.g., Brouwer's Theorem, James's Theorem -- are included when they are sufficiently important for the themes of the book.

As much as possible, I have tried to make this book current. Most mathematical papers published each year are on advanced and specialized material, not appropriate for an introductory work. Only occasionally does a paper strengthen, simplify, or clarify some basic, classical ideas. I have combed the literature for these insightful papers as well as I could, but some of them are not well known; that is evident from their infrequent mentions in the Science Citation Index. Following are a few of HAF's unusual features:

• A thorough introduction to filters in Chapters 5 and 6, and nets in Chapter 7. Those tools are used extensively in later chapters. Included are ideas of Aarnes and Andenaes [1972] on the interchangeability of subnets and superfilters, making available the advantages of both theories of convergence. Also included, in 15.10, is Gherman's [1980] characterization of topological convergences, which simplifies slightly the classic characterization of Kelley [1955/1975].
• an introduction to symmetric and preregular spaces, filling the conceptual gaps that are left in most introductions to T0, T1, T2, and T3 spaces -- see the table in 16.1.
• a unified treatment of topological spaces, uniform spaces, topological Abelian groups, topological vector spaces, locally convex spaces, Frechet spaces, Banach spaces, and Banach lattices, explaining these spaces in terms of increasingly specialized kinds of "distances" -- see the table in 26.1.
• converses to Banach's Contraction Fixed Point Theorem, due to Bessaga [1959] and Meyers [1967], in Chapter 19. These converses show that, although Banach's theorem is quite easy to prove, a longer proof cannot yield stronger results.
• the Brouwer Fixed Point Theorem, proved via van Maaren's geometry-free version of Sperner's Lemma. This approach is particularly intuitive and elementary in that it involves neither Jacobians nor triangulations. It decomposes the proof of Brouwer's Theorem into a purely combinatorial argument (in 3.28) and a compactness argument (in 27.19).
• introductions to both the Lebesgue and Henstock integrals and a proof of their equivalence in Chapter 24. (More precisely, a Banach- space- valued function is Lebesgue integrable if and only if it is almost separably valued and absolutely Henstock integrable.)
• pathological examples due to Nedoma, Kottman, Gordon, Dieudonne, and others, which illustrate very vividly some of the differences between Rn and infinite-dimensional Banach spaces.
• an introduction to set theory, including the most interesting equivalents of the Axiom of Choice, Dependent Choice, the Ultrafilter Principle, and the Hahn- Banach Theorem. (For lists of equivalents of these principles, see the index.)
• an introduction to formal logic following the substitution rules of Rasiowa and Sikorski [1963], which are simpler and -- in this author's opinion -- more natural than the substitution rules used in most logic textbooks. This is discussed in 14.20.
• a discussion of model theory and consistency results, including a summary of some results of Solovay, Pincus, Shelah, et al. Those results can be used to prove the nonconstructibility of many classical pathological objects of analysis; see especially the discussions in 14.76 and 14.77.
• Neumann's [1985] nonlinear Closed Graph Theorem.
• the automatic continuity theorems of Garnir [1974] and Wright [1977]. These results are similar to Neumann's, but instead of assuming a closed graph, they replace conventional set theory with ZF + DC + BP. Their result explains in part why a Banach space in applied math has a "usual norm;" see 14.77.

In compiling this book I have acted primarily as a reporter, not an inventor or discoverer. Nearly all the theorems and proofs in HAF can be found in earlier books or in research journal articles -- but in many cases those books or articles are hard to find or hard to read. This book's goal is to enhance classical results by modernizing the exposition, arranging separate topics into a unified whole, and occasionally incorporating some recent developments.

I have tried to give credit where it is due, but that is sometimes difficult or impossible. Historical inaccuracies tend to propagate through the literature. I have tried to weed out the inaccuracies by reading widely, but I'm sure I have not caught them all. Moreover, I have not always distinguished between primary and secondary sources. In many cases I have cited a textbook or other secondary source, to give credit for an exposition that I have modified in the present work.

## Existence, Examples, and Intangibles

Most existence proofs use either compactness, completeness, or the Axiom of Choice; those topics receive extra attention in this book. (In fact, Choice, Completeness, Compactness was the title of an earlier, prepublication version of this book; papers that mention that title are actually citing this book.) Although those three approaches to existence are usually quite different, they are not entirely unrelated -- AC has many equivalent forms, some of which are concerned with compactness or completeness (see 17.16 and 19.13).

The term "foundations" has two meanings; both are intended in the title of this book:

(i) In nonmathematical, everyday English, "foundations" refers to any basic or elementary or prerequisite material. For instance, this book contains much elementary set theory, algebra, and topology. Those subjects are not part of analysis, but are prerequisites for some parts of analysis.

(ii) "Foundations" also has a more specialized and technical meaning. It refers to more advanced topics in set theory (such as the Axiom of Choice) and to formal logic. Many mathematicians consider these topics to be the basis for all of mathematics.

Conventional analysis books include only a page or so concerning (ii); this book contains much more. We are led to (ii) when we look for examples of pathological objects.

Students and researchers need examples; it is a basic precept of pedagogy that every abstract idea should be accompanied by one or more concrete examples. Therefore, when I began writing this book (originally a conventional analysis book), I resolved to give examples of everything. However, as I searched through the literature, I was unable to find explicit examples of several important pathological objects, which I now call intangibles:

• finitely additive probabilities that are not countably additive,
• elements of (a customary corollary of the Hahn- Banach Theorem),
• universal nets that are not eventually constant,
• free ultrafilters (used very freely in nonstandard analysis!),
• well orderings for R,
• inequivalent complete norms on a vector space,

etc. In analysis books it has been customary to prove the existence of these and other pathological objects without constructing any explicit examples, without explaining the omission of examples, and without even mentioning that anything has been omitted. Typically, the student does not consciously notice the omission, but is left with a vague uneasiness about these unillustrated objects that are so difficult to visualize.

I could not understand the dearth of examples until I accidentally ventured beyond the traditional confines of analysis. I was surprised to learn that the examples of these mysterious objects are omitted from the literature because they must be omitted: Although the objects exist, it can also be proved that explicit constructions do not exist. That may sound paradoxical, but it merely reflects a peculiarity in our language: The customary requirements for an "explicit construction" are more stringent than the customary requirements for an "existence proof." In an existence proof we are permitted to postulate arbitrary choices, but in an explicit construction we are expected to make choices in an algorithmic fashion. (To make this observation more precise requires some definitions, which are given in 14.76 and 14.77.)

Though existence without examples has puzzled some analysts, the relevant concepts have been a part of logic for many years. The nonconstructive nature of the Axiom of Choice was controversial when set theory was born about a century ago, but our understanding and acceptance of it has gradually grown. An account of its history is given by Moore [1982]. It is now easy to observe that nonconstructive techniques are used in many of the classical existence proofs for pathological objects of analysis. It can also be shown, though less easily, that many of those existence theorems cannot be proved by other, constructive techniques. Thus, the pathological objects in question are inherently unconstructible.

The paradox of existence without examples has become a part of the logicians' folklore, which is not easily accessible to nonlogicians. Most modern books and papers on logic are written in a specialized, technical language that is unfamiliar and nonintuitive to outsiders: Symbols are used where other mathematicians are accustomed to seeing words, and distinctions are made which other mathematicians are accustomed to blurring -- e.g., the distinction between first-order and higher-order languages. Moreover, those books and papers of logic generally do not focus on the intangibles of analysis.

On the other hand, analysis books and papers invoke nonconstructive principles like magical incantations, without much accompanying explanation and -- in some cases -- without much understanding. One recent analysis book asserts that analysts would gain little from questioning the Axiom of Choice. I disagree. The present work was motivated in part by my feeling that students deserve a more "honest" explanation of some of the non-examples of analysis -- especially of some of the consequences of the Hahn- Banach Theorem. When we cannot construct an explicit example, we should say so. The student who cannot visualize some object should be reassured that no one else can visualize it either. Because examples are so important in the learning process, the lack of examples should be discussed at least briefly when that lack is first encountered; it should not be postponed until some more advanced course or ignored altogether.

Though most of HAF relies only on conventional reasoning -- i.e., the kind of set theory and logic that most mathematicians use without noticing they are using it -- we come to a better understanding of the idiosyncrasies of conventional reasoning by contrasting it with unconventional systems, such as ZF + DC + BP or Bishop's constructivism. HAF explains the relevant foundational concepts in brief, informal, intuitive terms that should be easily understood by analysts and other nonlogicians.

To better understand the role played by the Axiom of Choice, we shall keep track of its uses and the uses of certain weakened forms of AC, especially

the Principle of Dependent Choices (DC), which is constructive and is equivalent to several principles about complete metric spaces;

the Ultrafilter Principle (UF), which is nonconstructive and is equivalent to the Completeness and Compactness Principles of logic, as well as dozens of other important principles involving topological compactness; and

the Hahn- Banach Theorem (HB), also nonconstructive, which has many important equivalent forms in functional analysis.

Most analysts are not accustomed to viewing HB as a weakened form of AC, but that viewpoint makes the Hahn- Banach Theorem's nonconstructive nature much easier to understand.

This book's sketch of logic omits many proofs and even some definitions. It is intended not to make the reader into a logician, but only to show analysts the relevance of some parts of logic. The introduction to foundations for analysts is HAF's most unusual feature, but it is not an overriding feature -- it takes up only a small portion of the book and can be skipped over by mathematicians who have picked up this book for its treatment of nonfoundational topics such as nets, F-spaces, or integration.

## Abstract versus Concrete

I have attempted to present each set of ideas at a natural level of generality and abstraction -- i.e., a level that conveys the ideas in a simple form and permits several examples and applications. Of course, the level of generality of any part of the book is partly dictated by the needs of later parts of the book.

Usually, I lean toward more abstract and general approaches when they are available. By omitting unnecessary, irrelevant, or distracting hypotheses, we trim a concept down to reveal its essential parts. In many cases, omitting unnecessary hypotheses does not lengthen a proof, and it may make the proof easier to understand because the reader's attention is then focused on the few possible lines of reasoning that still remain available. For instance, every metric space can be embedded isometrically in a Banach space (see 22.14), but the "more concrete" setting of Banach spaces does not improve our understanding of metric space results such as the Contraction Fixed Point Theorem in 19.39.

Here is another example of my preference for abstraction: Some textbooks build Hausdorffness into their definition of "uniform space" or "topological vector space" or "locally convex space" because most spaces used in applications are in fact Hausdorff. This may shorten the statements of several theorems by a word or two, but it does not shorten the proofs of those theorems. Moreover, it may confuse beginners by entangling concepts that are not inherently related: The basic ideas of Hausdorff spaces are independent from the other basic ideas of uniform spaces, topological spaces, and locally convex spaces; neither set of ideas actually requires the other. In HAF, Hausdorffness is a separate property; it is not built into our definitions of those other spaces. Our not-necessarily- Hausdorff approach has several benefits, of which the greatest probably is this:

The weak topology of an infinite-dimensional Banach space is an important nonmetrizable Hausdorff topology that is best explained as the supremum of a collection of pseudometrizable, non- Hausdorff topologies.

(If the reader is accustomed to working only in Hausdorff spaces, HAF's not-necessarily- Hausdorff approach may take a little getting used to, but only a little. Mostly, one replaces "metric" with "pseudometric" or with the neutral notion of "distance;" one replaces "the limit" with "a limit" or with the neutral notion of "converges to.")

However, a more general approach to a topic is not necessarily a simpler approach. Every idea in mathematics can be made more general and more abstract by making the hypotheses weaker and more complicated and by introducing more definitions, but I have tried to avoid the weakly upper hemisemidemicontinuous quasipseudospaces of baroque mathematics. It is unavoidable that the beginning graduate student of mathematics must wade through a large collection of new definitions, but that collection should not be made larger than necessary. Thus we sometimes accept slightly stronger hypotheses for a theorem in order to avoid introducing more definitions. Of course, ultimately the difference between important distinctions and excessive hair-splitting is a matter of an individual mathematician's own personal taste.

Converses to main implications are included in HAF whenever this can be managed conveniently, as well as in a few inconvenient cases that I deemed sufficiently important. Lists of dissimilar but equivalent definitions are collected into one long definintion-and-theorem, even though that one theorem may have a painfully long proof. The single portmanteau theorem is convenient for reference, and moreover it clearly displays the importance of a concept. For instance, the notion of "ultrabarrelled spaces" seemed too advanced and specialized for this book until I saw the long list of dissimilar but equivalent definitions that now appears in 27.26. To prevent confusion, I have called the student's attention to contrasts between similar but inequivalent concepts, either by juxtaposing them (as in the case of barrels and ultrabarrels) or by including cross-referencing remarks (as in the case of Bishop's constructivism and Godel's constructivism).

Although the content is chosen for analysts, the writing style has been influenced by algebraists. Whenever possible, I have made degenerate objects such as the empty set into a special case of a rule, rather than an exception to the rule. For instance, in this book and in algebra books, is an "improper filter" on X, though it is not a filter at all according to the definition used by many books on general topology.

## Order of Topics

I have followed a Bourbaki-like order of topics, first introducing simple fundamentals and later building upon them to develop more specialized ideas. The topics are ordered to suit pedagogy rather than to emphasize applications. For instance, convexity is commonly introduced in functional analysis courses in the setting of Banach spaces or topological vector spaces, but I have found it expedient to introduce convexity as a purely algebraic notion, and then add topological considerations much later in the book. Most topological vector spaces used in applications are locally convex, but HAF first studies topological vector spaces without the additional assumption of local convexity.

Topics covered within a single chapter are closely related to each other. However, in many cases the end of a chapter covers more advanced and specialized material that can be postponed; it will not be needed until much later in the book, if at all. Most of Part C (on topological and uniform spaces) can be read without Part B (logic and algebra). However, most readers should skim through Chapters 5, 6, and 7. Those chapters introduce filters and nets -- tools that are used more extensively in this book than in most analysis books.

I have felt justified in violating logical sequencing in one important instance. The real number system is, in some sense, the foundation of analysis, so it must be used in examples quite early in the book. Examples given in early chapters assume an informal understanding of the real numbers, such as might be acquired in calculus and other early undergraduate courses. A more precise definition of the reals is neither needed nor attainable until Chapter 10. Much conceptual machinery must be built before we can understand and prove a statement such as this one:

There exists a Dedekind complete, chain ordered field, called the real numbers. It is unique up to isomorphism if we use the conventional reasoning methods of analysts. (It is not unique if we restrict our reasoning methods to first-order languages and permit the use of nonstandard models.)

The existence and uniqueness of the complete ordered field justify the usual definition of R. I am surprised that these algebraic results are not proved (or even mentioned!) in many introductory textbooks on analysis.

A traditional course on measure and integration would correspond roughly to part of Chapter 11, all of Chapter 21, and parts of Chapters 22-25 and 29. Integration theory is commonly introduced separately from functional analysis, but I have mixed the two topics together because I feel that each supports the other in essential ways. All of the usual definitions of the Lebesgue space L1[0,1] (e.g., in 19.38, 22.28, or 24.36) are quite involved; these definitions cannot be properly appreciated without some of the abstract theory of completions or Banach spaces or convergent nets. Conversely, an introduction to Banach spaces is narrow or distorted if it omits or postpones the rather important example of Lp spaces; the remaining elementary examples of Banach spaces are not diverse enough to give a proper feel for the subject.

## How to Use This Book

Because students' backgrounds differ greatly, I have tried to assume very few prerequisites. The book is intended for students who have finished calculus plus at least four other college math courses. HAF will rely on those four additional courses, not for specific content, but only for mathematical maturity -- i.e., for the student's ability to learn new material at a certain pace and a certain level of abstraction, and to fill in a few omitted details to make an exercise into a proof. Students with that amount of preparation will find HAF self-contained; they will not need to refer to other books to read this one. Students with sufficient mathematical maturity may not even need to refer to their college calculus textbooks; Chapters 24 and 25 reintroduce calculus in the more general setting of Banach spaces. Proofs are included, or at least sketched, for all the main results of this book except a few consistency results of formal logic. For those consistency results we give references in lieu of proofs, but the conclusions are explained in sufficient detail to make them clear to beginners.

Parts of HAF might be used as a classroom textbook, but HAF was written primarily for individual use. My intended reader will skip back and forth from one part of the book to another; different readers will follow different paths through the book. The reader should begin by skimming the table of contents to get acquainted with the ordering of topics. To facilitate skipping around in the book, I have included a large index and many cross-referencing remarks. Newly defined terms are generally given in boldface to make them easy to find. These definitions are followed by alternate terminology in italics if the literature uses other terms for the same concept or by cautionary remarks if the literature also uses the same term for other concepts. The first few pages of the first chapter introduce many of the symbols and typographical conventions used throughout the book; the index ends with a list of symbols. A list of charts, tables, diagrams, and figures is included in the index under "charts."

Mathematics textbooks usually postpone exercises until the end of each subchapter or each chapter, but HAF mixes exercises into the main text. In fact, HAF does not always distinguish sharply between "discussions," "theorems," "examples," and "exercises." All such assertions are true statements, with varying degrees of importance, generality, or difficulty, and with varying amounts of hints provided. Every student knows that reading through any proof in any math book is a challenge, whether that proof is marked "exercise" or not. Some computations and deductions are easier or more instructive to do than to watch, so for brevity I have intentionally given some proofs as sketches. All the "exercises" are actually part of the text; most of them will serve as essential examples or as steps in proofs of later theorems. Thus, in each chapter that is studied, the reader should work through, or at least READ through, every exercise; no exercise should be skipped.

## Acknowledgments

I am especially grateful to Isidore Fleischer, Mai Gehrke, Paul Howard, and Constantine Tsinakis, who helped with innumerable questions about algebra and logic. I am also grateful to many other mathematicians who helped or tried to help with many different questions: Richard Ball, Howard Becker, Lamar Bentley, Dan Biles, Andreas Blass, Douglas Bridges, Norbert Brunner, Gerard Buskes, Chris Ciesielski, John Cook, Matthew Foreman, Doug Hardin, Peter Johnstone, Bjarni Jonsson, William Julian, Keith Kearnes, Darrell Kent, Menachem Kojman, Ralph Kopperman, Wilhelmus Luxemburg, Hans van Maaren, Roman Manka, Peter Massopust, Ralph McKenzie, Charles Megibben, Norm Megill, Michael Mihalik, Zuhair Nashed, Neil Nelson, Michael Neumann, Jeffrey Norden, Simeon Reich, Fred Richman, Saharon Shelah, Stephen Simons, Steve Tschantz, Stan Wagon, and others too numerous to list here. I am also grateful to many students who read through earlier versions of parts of this book. Of course, any mistakes that remain in this book are my own.

This work was supported in part by a Summer Award from the Vanderbilt University Research Council. I would also like to thank John Cook, Mark Ellingham, Martin Fryd, Bob Messer, Ruby Moore, Steve Tschantz, John Williams, and others for their help with TeX. This book was composed using several different computers and wordprocessors. It was typeset using LaTeX, with some fonts and symbols imported from AmSTeX.

I am also grateful to my family for their support of this project.

## To Contact Me

I've surveyed the literature as well as I could, but it's enormous; I'm sure there is much that I've overlooked. I would be grateful for comments from readers, particularly regarding errors or other suggested alterations for a possible later edition. I will post the errata and other insights on the book's World Wide Web page on the internet.

Eric Schechter, August 16, 1996
http://math.vanderbilt.edu/~schectex/ccc/