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This chapter, beginning with Section 2, develops the topic of sequences and series of functions, especially of functions of one variable. An important part of the treatment is an introduction to the problem of interchange of limits, both theoretically and practically. This problem plays a role repeatedly in real analysis, but its visibility decreases as more and more results are developed for handling it in various situations. Fourier series are introduced in this chapter and are carried along throughout the book as a motivating example for a number of problems in real analysis.

Section 1 makes contact with the core of a first undergraduate course in real-variable theory. Some material from such a course is repeated here in order to establish notation and a point of view. Omitted material is summarized at the end of the section, and some of it is discussed in a little more detail in an appendix at the end of the book. The point of view being established is the use of defining properties of the real number system to prove the Bolzano-Weierstrass Theorem, followed by the use of that theorem to prove some of the difficult theorems that are usually assumed in a one-variable calculus course. The treatment makes use of the extended real-number system, in order to allow sup and inf to be defined for any nonempty set of reals and to allow lim sup and lim inf to be meaningful for any sequence.

Sections 2–3 introduce the problem of interchange of limits. They show how certain concrete problems can be viewed in this way, and they give a way of thinking about all such interchanges in a common framework. A positive result affirms such an interchange under suitable hypotheses of monotonicity. This is by way of introduction to the topic in Section 3 of uniform convergence and the role of uniform convergence in continuity and differentiation.

Section 4 gives a careful development of the Riemann integral for real-valued functions of one variable, establishing existence of Riemann integrals for bounded functions that are discontinuous at only finitely many points, basic properties of the integral, the Fundamental Theorem of Calculus for continuous integrands, the change-of-variables formula, and other results. Section 5 examines complex-valued functions, pointing out the extent to which the results for real-valued functions in the first four sections extend to complex-valued functions.

Section 6 is a short treatment of the version of Taylor’s Theorem in which the remainder is given by an integral. Section 7 takes up power series and uses them to define the elementary transcendental functions and establish their properties. The power series expansion of (1 +x) for arbitrary complex p is studied carefully. Section 8 introduces Cesàro and Abel summability, which play a role in the subject of Fourier series. A converse theorem to Abel’s theorem is used to exhibit the function |x|as the uniform limit of polynomials on [- 1, 1]. The Weierstrass Approximation Theorem of Section 9 generalizes this example and establishes that every continuous complex-valued function on a closed bounded interval is the uniform limit of polynomials.

Section 10 introduces Fourier series in one variable in the context of the Riemann integral. The main theorems of the section are a convergence result for continuously differentiable functions, Bessel’s inequality, the Riemann-Lebesgue Lemma, Fejér’s Theorem, and Parseval’s Theorem.

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples.

Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions.

Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about ℝ.

Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences.

Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.”

Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine-Borel Theorem says that the compact sets of ℝ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano-Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.”

Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem.

Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous.

Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli’s Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone-Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable.

Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

This chapter gives a rigorous treatment of parts of the calculus of several variables.

Sections 1–3 handle the more elementary parts of the differential calculus. Section 1 introduces an operator norm that makes the space of linear functions from ℝ to ℝ or from ℂ to ℂ into a metric space. Section 2 goes through the definitions and elementary facts about differentiation in several variables in terms of linear transformations and matrices. The chain rule and Taylor’s Theorem with integral remainder are two of the results of the section. Section 3 supplements Section 2 in order to allow vector-valued and complex-valued extensions of all the results.

Sections 4–5 are digressions. The material in these sections uses the techniques of the present chapter but is not needed until later. Section 4 develops the exponential function on complex square matrices and establishes its properties; it will be applied in Chapter IV. Section 5 establishes the existence of partitions of unity in Euclidean space; this result will be applied at the end of Section 10.

Section 6 returns to the development in Section 2 and proves two important theorems about differential calculus. The Inverse Function Theorem gives sufficient conditions under which a differentiable function from an open set in ℝ into ℝ has a locally defined differentiable inverse, and the Implicit Function Theorem gives sufficient conditions for the local solvability of m nonlinear equations in n + m variables for m of the variables in terms of the other n. The Inverse Function Theorem is proved on its own, and the Implicit Function Theorem is derived from it.

Sections 7–10 treat Riemann integration in several variables. Elementary properties analogous to those in the one-variable case are in Section 7, a useful necessary and sufficient condition for Riemann integrability is established in Section 8, Fubini’s Theorem for interchanging the order of integration is in Section 9, and a preliminary change-of-variables theorem for multiple integrals is in Section 10.

This chapter treats the theory of ordinary differential equations, both linear and nonlinear.

Sections 1–4 establish existence and uniqueness theorems for ordinary differential equations. The first section gives some examples of first-order equations, mostly nonlinear, to illustrate certain kinds of behavior of solutions. The second section shows, in the presence of continuity for a vectorvalued F satisfying a “Lipschitz condition,” that the first-order system y′ = F(t, y) has a unique local solution satisfying an initial condition y (t) = y. Since higher-order equations can always be reduced to first-order systems, these results address existence and uniqueness for n-order equations as a special case. Section 3 shows that the solutions to a system depend well on the initial condition and on any parameters that are present in F. Section 4 applies these results to existence of integral curves for a vector field and to construction of coordinate systems from families of integral curves.

Sections 5–8 concern linear systems. Section 5 shows that local solutions of linear systems may be extended to global solutions and that in the homogeneous case the vector space of global solutions has dimension equal to the size of the system. The method of variation of parameters reduces the solution of any linear system to the solution of a homogeneous linear system. Sections 6–7 identify explicit solutions to n-order linear equations and first-order linear systems. The “Jordan canonical form” of a square matrix plays a role in the case of a system. Section 8 discusses power-series solutions to second-order homogeneous linear equations whose coefficients are given by convergent power series, as well as solutions that arise in the case of regular singular points. Two kinds of special functions are mentioned that result from this study-Legendre polynomials and Bessel functions.

This chapter develops the basic theory of measure and integration, including Lebesgue measure and Lebesgue integration for the line.

Section 1 introduces measures, including 1-dimensional Lebesgue measure as the primary example, and develops simple properties of them. Sections 2–4 introduce measurable functions and the Lebesgue integral and go on to establish some easy properties of integration and the fundamental theorems about how Lebesgue integration behaves under limit operations.

Sections 5–6 concern the Extension Theorem announced in Section 1 and used as the final step in the construction of Lebesgue measure. The theorem allows σ-finite measures to be extended from algebras of sets to σalgebras. The theorem is proved in Section 5, and the completion of a measure space is defined in Section 6 and related to the proof of the Extension Theorem.

Section 7 treats Fubini’s Theorem, which allows interchange of order of integration under rather general circumstances. This is a deep result. As part of the proof, product measure is constructed and important measurability conditions are established. This section mentions that Fubini’s Theorem will be applicable to higher-dimensional Lebesgue measure, but the details are deferred to Chapter VI.

Section 8 extends Lebesgue integration to complex-valued functions and to functions with values in finite-dimensional vector spaces.

Section 9 gives a careful definition of the spaces L, L, and L for any measure space, introduces the notion of a normed linear space, and verifies that these three spaces are examples. The main theorem of the section about L, L, and L is the completeness of these three spaces as metric spaces. In addition, the section proves a version of Alaoglu’s Theorem concerning weak-star convergence.

This chapter mines some of the powerful consequences of the basic measure theory in Chapter V.

Sections 1–3 establish properties of Lebesgue measure and other Borel measures on Euclidean space and on open subsets of Euclidean space. The main general property is the regularity of all such measures—that the measure of any Borel set can be approximated by the measure of compact sets from within and open sets from without. Lebesgue measure in all of Euclidean space has an additional property, translation invariance, which allows for the notion of the convolution of two functions. Convolution gives a kind of moving average of the translates of one function weighted by the other function. Convolution with the dilates of a fixed integrable function provides a handy kind of approximate identity.

Section 4 gives the final form of the comparison of the Riemann and Lebesgue integrals, a preliminary form having been given in Chapter III.

Section 5 gives the final form of the change-of-variables theorem for integration, starting from the preliminary form of the theorem in Chapter III and taking advantage of the ease with which limits can be handled by the Lebesgue integral. Sard’s Theorem allows one to disregard sets of lower dimension in establishing such changes of variables, thereby giving results in their expected form rather than in a form dictated by technicalities.

Section 6 concerns the Hardy-Littlewood Maximal Theorem in dimensions. In dimension 1, this theorem implies that the derivative of a 1 -dimensional Lebesgue integral with respect to Lebesgue measure recovers the integrand almost everywhere. The theorem in the general case implies that certain averages of a function over small sets about a point tend to the function almost everywhere. But the theorem can be regarded as saying also that a particular approximate identity formed by dilations applies to problems of almost-every where convergence, as well as to problems of norm convergence and uniform convergence. A corollary of the theorem is that many approximate identities formed by dilations yield almost-everywhere convergence theorems.

Section 7 redevelops the beginnings of the subject of Fourier series using the Lebesgue integral, the theory having been developed with the Riemann integral in Section I.10. With the Lebesgue integral and its accompanying tools, Fourier series are meaningful for more functions than before, Dini’s test applies even to a wider class of Riemann integrable functions than before, and Fejér’s Theorem and Parseval’s Theorem become easier and more general than before. A completely new result with the Lebesgue integral is the Riesz-Fischer Theorem, which characterizes the trigonometric series that are Fourier series of square-integrable functions.

Sections 8–10 deal with Stieltjes measures, which are Borel measures on the line, and their application to Fourier series. Such measures are characterized in terms of a class of monotone functions on the line, and they lead to a handy generalization of the integration-by-parts formula. This formula allows one to bound the size of the Fourier coefficients of functions of bounded variation, which are differences of monotone functions. In combination with earlier results, this bound yields the Dirichlet-Jordan Theorem, which says that the Fourier series of a function of bounded variation converges pointwise everywhere, the convergence being uniform on any compact set on which the function is continuous. Section 10 is a short section on computation of integrals.

This chapter concerns the Fundamental Theorem of Calculus for the Lebesgue integral, viewed from Lebesgue’s perspective but slightly updated.

Section 1 contains Lebesgue’s main tool, a theorem saying that monotone functions on the line are differentiable almost everywhere. A relatively easy consequence is Fubini’s theorem that an absolutely convergent series of monotone increasing functions may be differentiated term by term. The result that the indefinite integral of a locally integrable function is differentiable almost everywhere with derivative follows readily.

Section 2 addresses the converse question of what functions have the property for a particular that the integral can be evaluated all and The development involves a decomposition theorem for monotone increasing functions and a corresponding decomposition theorem for Stieltjes measures. The answer to the converse question when >- 0 and almost everywhere is that is “absolutely continuous” in a sense defined in the section.

This chapter develops some of the theory of the ℝ Fourier transform as an operator that carries certain spaces of complex-valued functions on ℝ to other spaces of such functions.

Sections 1–3 give the indispensable parts of the theory, beginning in Section 1 with the definition, the fact that integrable functions are mapped to bounded continuous functions, and various transformation rules. In Section 2 the main results concern L, chiefly the vanishing of the Fourier transforms of integrable functions at infinity, the fact that the Fourier transform is one-one, and the all-important Fourier inversion formula. The third section builds on these results to establish a theory for L. The Fourier transform carries functions in L ∩ L to functions in L, preserving the L norm; this is the Plancherel formula. The Fourier transform therefore extends by continuity to all of L, and the Riesz-Fischer Theorem says that this extended mapping is onto L. These results allow one to construct bounded linear operators on L commuting with translations by multiplying by L functions on the Fourier transform side and then using Fourier inversion; a converse theorem is proved in the next section.

Section 4 discusses the Fourier transform on the Schwartz space, the subspace of L consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. The Fourier transform carries the Schwartz space in one-one fashion onto itself, and this fact leads to the proof of the converse theorem mentioned above.

Section 5 applies the Schwartz space in ℝ to obtain the Poisson Summation Formula, which relates Fourier series and the Fourier transform. A particular instance of this formula allows one to prove the functional equation of the Riemann zeta function.

Section 6 develops the Poisson integral formula, which transforms functions on ℝ into harmonic functions on a half space in ℝ. A function on ℝ can be recovered as boundary values of its Poisson integral in various ways.

Section 7 specializes the theory of the previous section to ℝ, where one can associate a “conjugate” harmonic function to any harmonic function in the upper half plane. There is an associated conjugate Poisson kernel that maps a boundary function to a harmonic function conjugate to the Poisson integral. The boundary values of the harmonic function and its conjugate are related by the Hilbert transform, which implements a “90° phase shift” on functions. The Hilbert transform is a bounded linear operator on L and is of weak type (1, 1).

This chapter extends the theory of the spaces L, L, and L to include a whole family of spaces , 1 <- , in order to be able to capture finer quantitative facts about the size of measurable functions and the effect of linear operators on such functions.

Sections 1–2 give the basics about . For general measure spaces these consist of Hölder’s inequality, Minkowski’s inequality, a completeness theorem, and related results. For Euclidean space they include also facts about convolution.

Sections 3–4 develop some tools that at first may seem quite unrelated to spaces but play a significant role in Section 5. These are the Radon-Nikodym Theorem and two decomposition theorems for additive set functions. The Radon-Nikodym Theorem gives a sufficient condition for writing a measure as a function times another measure.

Section 5 identifies the space of continuous linear functionals on for 1 <- < ∞ when the underlying measure is σ-finite. For one thing this identification makes Alaoglu’s Theorem in Chapter V concrete enough so as to be quite useful.

Section 6 discusses the Marcinkiewicz Interpolation Theorem, which allows one to reinterpret suitably bounded operators between two pairs of spaces as bounded between intermediate pairs of spaces as well. The theorem has immediate corollaries for the Hardy-Littlewood maximal function and an approximation to the Hilbert transform, and Section 6 goes on to use each of these corollaries to derive interesting consequences.

This chapter extends considerably the framework for discussing convergence, limits, and continuity that was developed in Chapter II: topological spaces replace metric spaces.

Section 1 makes various definitions, including definitions for the terms topology, open set, closed set, continuous function, base for a topology, separable, and subspace. It introduces two general kinds of constructions useful in analysis and other fields for forming new topological spaces out of old ones—weak topologies and quotient topologies. The section gives several examples of each.

Sections 2–3 develop standard facts, mostly elementary, about how certain combinations of properties of topological spaces imply others. Examples show some limitations to such implications. Properties that are studied include Hausdorff, regular, normal, dense, compact, locally compact, Lindelöf, and σ-compact.

Section 4 discusses product topologies on arbitrary product spaces, an example of a weak topology. The main theorem, the Tychonoff Product Theorem, says that the product of compact spaces is compact.

Section 5 introduces nets, a generalization of sequences. Sequences by themselves are inadequate for detecting convergence in general topological spaces, and nets are a substitute. The use of nets in many cases provides an easier way of establishing properties of subsets of a topological space than direct arguments with open and closed sets.

Section 6 elaborates on quotient topologies as introduced in Section 1. Conditions under which a quotient space is Hausdorff are of particular interest.

Sections 7–8 prove and apply Urysohn’s Lemma, which says that any two disjoint closed sets in a normal topological space may be separated by a real-valued continuous function. This result is fundamental to serious uses of topological spaces in analysis. One application is to showing that every separable Hausdorff regular topology arises from a metric.

Section 9 extends Ascoli’s Theorem and the Stone-Weierstrass Theorem from their settings in compact metric spaces in Chapter II to the wider setting of compact Hausdorff spaces.