Functional analysis is concerned with infinite-dimensional linear spaces, such as Banach spaces and Hilbert spaces, which most often consist of functions or equivalence classes of functions. Each Banach space X has a dual space X′ defined as the set of all continuous linear functions from X into the field ℝ or ℂ.

One of the main examples of duality is for Lp spaces. Let (X, S, μ) be a measure space. Let 1 < p < ∞ and 1/p + 1/q = 1. Then it turns out that Lp and Lq are dual to each other via the linear functional f ↦ ∫ f g d μ for f in p and g in q. For p = q = 2, L2 is a Hilbert space, where it was shown previously that any continuous linear form on a Hilbert space H is given by inner product with a fixed element of H (Theorem 5.5.1).

Other than linear subspaces, some of the most natural and frequently applied subsets of a vector space S are the convex subsets C, such that for any x and y in C, and 0 < t < 1, we have tx + (1 – t)y ∈ C. These sets are treated in §§6.2 and 6.6. A function for which the region above its graph is convex is called a convex function. §6.3 deals with convex functions. Convex sets and functions are among the main subjects of modern real analysis.

In constructing a building, the builders may well use different techniques and materials to lay the foundation than they use in the rest of the building. Likewise, almost every field of mathematics can be built on a foundation of axiomatic set theory. This foundation is accepted by most logicians and mathematicians concerned with foundations, but only a minority of mathematicians have the time or inclination to learn axiomatic set theory in detail.

To make another analogy, higher-level computer languages and programs written in them are built on a foundation of computer hardware and systems programs. How much the people who write high-level programs need to know about the hardware and operating systems will depend on the problem at hand.

In modern real analysis, set-theoretic questions are somewhat more to the fore than they are in most work in algebra, complex analysis, geometry, and applied mathematics. A relatively recent line of development in real analysis, “nonstandard analysis,” allows, for example, positive numbers that are infinitely small but not zero. Nonstandard analysis depends even more heavily on the specifics of set theory than earlier developments in real analysis did.

This chapter will give only enough of an introduction to set theory to define some notation and concepts used in the rest of the book. In other words, this chapter presents mainly “naive” (as opposed to axiomatic) set theory.

General topology has to do with, among other things, notions of convergence. Given a sequence xn of points in a set X, convergence of xn to a point x can be defined in different ways. One of the main ways is by a metric, or distance d, which is nonnegative and real-valued, with xn → x meaning d(xn, x) → 0. The usual metric for real numbers is d(x, y) = ∣x - y∣. For the usual convergence of real numbers, a function f is called continuous if whenever xn → x in its domain, we have f(xn) → f(x).

On the other hand, some interesting kinds of convergence are not defined by metrics: if we define convergence of a sequence of functions fn “pointwise,” so that fn → f means fn(x) → f(x) for all x, it turns out that (for a large enough class of functions defined on an uncountable set) there may be no metric e such that fn→ f is equivalent to e(fn, f) → 0.

Given a sense of convergence, we can call a set F closed if whenever xi ∈ F for all i and xi → x we have x ∈ F also. Any closed interval [a, b] ≔ x: a ≤ x ≤ b is an example of a closed set. The properties of closed sets F and their complements U ≔ X∖F, which are called open sets, turn out to provide the best and most accepted way of extending the notions of convergence, continuity, and so forth to nonmetric situations.

Recall that a topological space is called locally compact iff every point in it is contained in some open set with compact closure. For some time, locally compact spaces were considered the most natural spaces on which to study general measures and integrals. The regularity extension, treated in §7.3, appeared as a primary advantage of measures on compact or locally compactspaces. Local compactness also fits well with group structures. A topological group is a group G with a Hausdorff topology  for which the group operations 〈g, h〉 ↦ gh and g ↦ g−1 are continuous. A locally compact Hausdorff topological group has left and right Haar measures, which are Radon measures (finite on compact sets) invariant under all left or right translations g ↦ hg or g ↦ gh, respectively. The theory of measures on locally compact spaces, particularly groups, occupied the last three of the twelve chapters of the classic text of Halmos (1950). Bourbaki (1952–1969) put even more emphasis on locally compact spaces. The main purpose of this appendix is to indicate why locally compact spaces have had less attention in this book than in those of Halmos and Bourbaki.

Another class of spaces for measures theory is complete separable metric spaces, or topological spaces which can be metrized to be separable and complete (Polish spaces). Among such spaces, for example, are separable, infinite-dimensional Banach spaces, none of which is locally compact.

This is a text at the beginning graduate level. Some study of intermediate analysis in Euclidean spaces will provide helpful background, but in this edition such background is not a formal prerequisite. Efforts to make the book more self-contained include inserting material on the real number system into Chapter 1, adding a treatment of the Stone-Weierstrass theorem, and generally eliminating references for proofs to other books except at very few points, such as some complex variable theory in Appendix B.

Chapters 1 through 5 provide a one-semester course in real analysis. Following that, a one-semester course on probability can be based on Chapters 8 through 10 and parts of 11 and 12. Starred paragraphs and sections, such as those found in Chapter 6 and most of Chapter 7, are called on rarely, if at all, later in the book. They can be skipped, at least on first reading, or until needed.

Relatively few proofs of less vital facts have been left to the reader. I would be very glad to know of any substantial unintentional gaps or errors. Although I have worked and checked all the problems and hints, experience suggests that mistakes in problems, and hints that may mislead, are less obvious than errors in the text. So take hints with a grain of salt and perhaps make a first try at the problems without using the hints.

Probabilities are easiest to define on finite sets. For example, consider a toss of a fair coin. Here “fair” means that heads and tails are equally likely. The situation may be represented by a set with two points H and T where H= “heads” and T= “tails.” The total probability of all possible outcomes is set equal to 1. Let “P(…)” denote “the probability of.…” If two possible outcomes cannot both happen, then one assumes that their probabilities add. Thus P(H) + P(T) = 1. By assumption P(H) = P(T), so P(H) = P(T)= 1/2.

Now suppose the coin is tossed twice. There are then four possible outcomes of the two tosses: HH, HT, TH, and TT. Considering these four as equally likely, they must each have probability 1/4. Likewise, if the coin is tossed n times, we have 2n possible strings of n letters H and T, where each string has probability 1/2n.

Next let n go to infinity. Then we have all possible infinite sequences of H's and T's. Each individual sequence has probability 0, but this does not determine the probabilities of other interesting sets of possible outcomes, as it did when n was finite. To consider such sets, first let us replace H by 1 and T by 0, precede the sequence by a “binary point” (as in decimal point), and regard the sequence as a binary expansion.

Borel Isomorphism

Two measurable spaces (X, ) and (Y, ) are called isomorphic iff there is a one-to-one function f from X onto Y such that f and f−1 are measurable. Two metric spaces (X, d) and (Y, e) will be called Borel-isomorphic, written X ∼ Y, iff they are isomorphic with their σ-algebras of Borel sets.

Clearly, Borel isomorphism comes somewhere between being homeomorphic topologically and being isomorphic as sets, which means having the same cardinality. The following main fact of this section shows that in many cases, surprisingly, Borel isomorphism is just equivalent to having the same cardinality:

Theorem If X and Y are two separable metric spaces which are Borel subsets of their completions, then X ∼ Y if and only if X and Y have the same cardinality, which moreover is either finite, countable, or c (the cardinal of the continuum, that is, of [0, 1]).

Remarks In general, the continuum hypothesis, stating that no sets have cardinality uncountable but strictly less than c, is independent of the other axioms of set theory, including the axiom of choice (see the notes to Appendix A.3). For Borel sets in complete separable metric spaces, however, the continuum hypothesis follows from the axioms, by the theorem about to be proved. Examples of the isomorphism are ℝ ∼ ℝ2 and ℝ ∼ ℝ∖, the space of irrational numbers.

The key idea of functional analysis is to consider functions as “points” in a space of functions. To begin with, consider bounded, measurable functions on a finite measure space (X, S, μ) such as the unit interval with Lebesgue measure. For any two such functions, f and g, we have a finite integral ∫ f g d μ = ∫ f(x)g(x)d μ(x). If we consider functions as vectors, then this integral has the properties of an inner product or dot product (f, g): it is nonnegative when f = g, symmetric in the sense that (f, g)≡ (g, f), and linear in f for fixed g. Using this inner product, one can develop an analogue of Euclidean geometry in a space of functions, with a distance d(f, g) = (f – g, f – g)½, just as in a finite-dimensional vector space. In fact, if μ is counting measure on a finite set with k elements, (f, g) becomes the usual inner product of vectors in ℝk. But if μ is Lebesgue measure on [0, 1], for example, then for the metric space of functions with distance d to be complete, we will need to include some unbounded functions f such that ∫ f2 dμ < ∞. Along the same lines, for each p > 0 and μ there is the collection of functions f which are measurable and for which ∫ |f|p d μ < ∞. This collection is called p or p(μ).