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Although it is common practice in calculus courses to discuss continuity before differentiation, historically mathematicians’ attention to the concept of continuity came long after the derivative was in wide use. Pierre de Fermat (1601–1665) was using tangent lines to solve optimization problems as early as 1629. On the other hand, it was not until around 1820 that Cauchy, Bolzano, Weierstrass, and others began to characterize continuity in terms more rigorous than prevailing intuitive notions such as “unbroken curves” or “functions which have no jumps or gaps.” The basic reason for this two-hundred year waiting period lies in the fact that, for most of this time, the very notion of function did not really permit discontinuities. Functions were entities such as polynomials, sines, and cosines, always smooth and continuous over their relevant domains. The gradual liberation of the term function to its modern understanding a rule associating a unique output to a given input—was simultaneous with 19th century investigations into the behavior of infinite series. Extensions of the power of calculus were intimately tied to the ability to represent a function f (x) as a limit of polynomials (called a power series) or as a limit of sums of sines and cosines (called a trigonometric or Fourier series). A typical question for Cauchy and his contemporaries was whether the continuity of the limiting polynomials or trigonometric functions necessarily implied that the limit f would also be continuous.

If we want to compute the value of a specific function at some point a, it may happen that we can compute only the values of the function near a. Consider, for example, the distance a free-falling object covers. This is given by the equation \(s(t) = g \cdot t^{2}/2\), where t is the time elapsed, and g is the gravitational constant. Knowing this equation, we can easily compute the value of s(t). If, however, we want to calculate s(t) at a particular time t = a by measuring the time, then we will not be able to calculate the precise distance corresponding to this given time; we will obtain only a better or worse approximation—depending on the precision of our instruments. However, if we are careful, we will hope that if we use the value of t that we get from the measurement to recover s(t), the result will be close to the original s(a). In essence, such difficulties always arise when we are trying to find some data with the help of another measured quantity. At those times, we assume that if our measured quantity differs from the real quantity by a very small amount, then the value computed from it will also be very close to its actual value.

In this chapter we consider a complete metric space of sequences of mappings acting on a bounded, closed and convex subset K of a Banach space which share a common convex Lyapunov function f. We introduce the concept of normality and show that a generic element taken from this space is normal. The sequence of values of the Lyapunov uniformly continuous function f along any (unrestricted) trajectory of such an element tends to the infimum of f on K. We establish a convergence result for perturbations of such trajectories. We then show that if f is Lipschitzian, then the complement of the set of normal sequences is σ-porous.

Roughly speaking, a quantity y is said to depend continuously on a quantity x if “small” changes in x result in small changes in y. Our goal in this chapter is to make this statement mathematically precise.

A metric space \(\mathrm{M} = (M;d )\) is homogeneous if for every isometry \(\alpha \) of a finite subspace of \(\mathrm{M}\) to a subspace of \(\mathrm{M}\) there exists an isometry of \(\mathrm{M}\) onto \(\mathrm{M}\) extending \(\alpha \). The metric space \(\mathrm{M}\) is universal if it isometrically embeds every finite metric space \(\mathrm{F}\) with \( dist (\mathrm{F}) \subseteq dist= (\mathrm{M})\). (\(dist (\mathrm{M})\) being the set of distances between points of \(\mathrm{M}\).) A metric space \(\mathrm{M}\) is oscillation stable if for every \(\epsilon > 0\) and every uniformly continuous and bounded function \(f : M \rightarrow \mathfrak{R}\) there exists an isometric copy \(\mathrm{{M}}^{{\ast}} = ({M}^{{\ast}};d )\) of \(\mathrm{M}\) in \(\mathrm{M}\) for which: $$\sup \{\vert f(x) - f(y)\vert \mid x,y \in {M}^{{\ast}}\} < \epsilon.$$

Consider a set of particles P 1, …, P n. To locate these particles in the space ℝ3, we need a reference system. Let the Cartesian coordinates of particle P i be (ξi, η i , ζ i ). Identifying (ξ1, η1, ξ1) with the triple (x1, x2, x 3), (ξ2, η2, ζ2) with (x,4, x,5, x,6), and so on, we obtain a vector x of the Euclidean space ℝ3n with coordinates (x1, x 2, …, x 3n ). Then x determines the positions of all particles in the set.

In this chapter, we consider a variety of Hilbert and Banach spaces that admit sampling expansions \(f(t) = \sum\nolimits_{n=1}^{\infty }f({t}_{n}){S}_{n}(t)\), where {S n } n=1 ∞ is a family of functions that depend on the sampling points {t n } n=1 ∞ but not on the function f. Those function spaces, that arise in connection with sampling expansions, include reproducing kernel spaces, Sobolev spaces, Wiener amalgam space, shift-invariant spaces, translation-invariant spaces, and spaces modeling signals with finite rate of innovation. Representative sampling theorems are presented for signals in each of these spaces. The chapter also includes recent results on nonlinear sampling of signals with finite rate of innovation, convolution sampling on Banach spaces, and certain foundational issues in sampling expansions.

In this chapter I deal with a frequently used method for constructing uniformities: Given a point-separating set of mappings from a set S to uniform spaces, one uniformity on S is the coarsest for which the mappings are uniformly continuous. This is the induced uniformity on S, also known as the projectively induced or projectively generated uniformity. Special cases of this construction are the uniform subspace, the uniform and semiuniform product and various “weak” uniform structures.

In this chapter I define uniform structures and uniformly continuous mappings in the language of pseudometrics. I derive their basic properties and their relationship to topologies.

Chapter 4 introduces the concept of p-th mean almost periodicity is introduced. It is shown that each p-th mean almost periodic stochastic process defined on a probability space (\(\Omega, \mathcal{F}, \mathbf{P}\)) is uniformly continuous and stochastically bounded. The collection of such stochastic processes is a Banach space when it is equipped with its natural norm. Moreover, two composition theorems for p-th mean almost periodic processes (Theorem 4.4 and Theorem 4.5) are established. They play a crucial role in the study of the existence (and uniqueness) of p-th mean almost periodic solutions to various stochastic differential equations on \(L^{P} (\Omega, \mathbb{H})\) where \(\mathbb{H}\) is a real separable Hilbert space.

Having considered the fundamental properties of the real numbers in Chapters 1 and 2, we now commence our study of real analysis proper. The heart of real analysis, and one of the key features that distinguishes real analysis from algebraic and combinatorial branches of mathematics, is the concept of a limit. There are various types of limits, for example limits of functions (which we will discuss in the present section), and limits of sequences (to be discussed in Section 8.2). However, all of these types of limits have similar features, and gaining familiarity with one type of limit will make learning about the other types much easier. The reader has already encountered limits of functions in an intuitive fashion in calculus courses. However, limits take on a much more important role in real analysis than in calculus because in the former we are concerned with rigorous proofs rather than applications, and limits are at the heart of the rigorous treatment of calculus.

It is likely that your first calculus class included a discussion of continuity. Many students find the definition hard to understand, and in many calculus classes the fine details are skipped. One of the rewards of mastering the kind of mathematics discussed in this book is that items like \(\varepsilon - \delta\) the definition of continuity are suddenly revealed as quite easy.

Let (X, α, P) be a probability space, x(1),x(2), ... coordinates for the countable product of copies of X, and ℱ ⊂ ℒ2 (X, α, P). Let vn be the normalized empirical measure n−1/2(δx(1)+...+δx(n)-nP). Let l∞(ℱ) be the set of all bounded real functions G on ℱ with norm ‖G‖ℱ := supfeℱ ¦G(f)¦. Let μ(f) := efdμ for any measure μ. We are interested in central limit theorems where vn converges in law for ‖·‖ℱ. The limit is a Gaussian process Gp with mean 0 and covariance $$EG_P (f)G_P (g) = P(fg) - P(f)P(g): = (f,g)_{P,O} ,$$ f,g e ℱ. Say ℱ is GPBUC if GP can be chosen to have each sample function bounded and uniformly continuous for (.,.)P,0. Say the central limit theorem holds for ℱ if ℱ is GPBUC and we have convergence of upper integrals limn→∞ ∝*H(vn)dpn = EH(GP) for every bounded continuous real function H on l∞(ℱ). (Thus, as noted by J. Hoffmann-Jorgensen, convergence "in law" does not require definition of laws.) In Sec. 4 an extended Wichura’s theorem is proved: given such convergence in law, there exist almost surely convergent realizations. Sec. 5 shows that the new definition of central limit theorem holding is equivalent to the previous definition of "functional Donsker class." Secs. 6–7 treat the case where ℱ={M1h≥M: 0

We survey results, obtained by the author and his coauthors over the last fifteen years, on optimal Sobolev embeddings and related inequalities in Orlicz spaces. Some of the presented results are very recent and are not published yet. We recall basic properties concerning Orlicz and Orlicz–Sobolev spaces and then dicsuss embeddings of Sobolev type and embeddings into spaces of uniformly continuous functions, classical and approximate differentiability properties of Orlicz–Sobolev functions and also trace inequalities on the boundary.

Current research on analysis in metric measure spaces has used alternative notions of Sobolev functions on metric measure spaces. We show that, under some mild geometric assumptions on the metric measure space, all these notions give the same class of functions.

We recall that a metric space is a pair (X, ρ), where X is a set and ρ:X × X → [0, ∞) is a metric, satisfying the following axioms: ρ(x, y) = 0 if and only if x = y, ρ(x, y) = ρ(y, x), and ρ(x, y) + ρ(y, z) ≥ ρ(x, z).

It is natural to think about distance between physical objects—people, say, or buildings or stars. In what follows, we explore — notion of “closeness” for such things as functions and sequences. (How far is f (x)= x3 from g(x) =sin x? How far is the sequence (1/n) from (2/n2)?) The way we answer such — question is through the idea of — metric space.In principle, it enables us to talk about the distance between colorsor ideas or songs. When we can measure “distance,” we can take limits or “perform analysis.” Special distance-measuring devices called norms are introduced for vector spaces. The analysis we care most about in this book involves norms. This type of analysis is known as functional analysis because the vector spaces of greatest interest are spaces of functions.

Given a function f, we wish to approximate f by functions having certain properties. There is a general method for doing this, which will now be described.

Intervals are convex (4.1.4) and closed intervals are compact (4.5.4); these special properties of intervals are reflected in special properties of continuous functions defined on them. Applications include the construction of n’th roots, a characterization of injectivity (§6.4) and automatic continuity of inverse functions (§6.5).

In this chapter, we study vector spaces over arbitrary fields that have a bilinear form defined on them. As we will see, the study of such vector spaces has a very geometric flavor, and hence so does the terminology.

We developed many of the basic properties of metric spaces in Chapter 6. A complete metric space was defined in Chapter 13 and we saw the importance of such spaces in the proof of the fundamental fixed point theorem (Theorem 13.2). This theorem, which has many applications, was used to prove the existence of solutions of ordinary differential equations and the Implicit function theorem. We now discuss in more detail functions whose domain is a metric space, and we prove convergence and approximation theorems which are useful throughout analysis.

The theory developed in Chapter IV justifies its existence by what it has to say about the limiting distributions of functional defined on sequences of stochastic processes. Processes X n (t):t ∈ T are identified with random elements of some space X of functions on T, a space large enough to contain the sample paths of every X n ; the functional are maps defined on X to which the Continuous Mapping Theorem can be applied. In this chapter we shall specialize the general theory to the particular function space D[0, 1], under its uniform metric. It will turn out that most applications, especially those that come up with brownian bridges and brownian motions as limit processes, require no fancier setting than this.

A metric space is a set equipped with a function which assigns a numerical “distance” to each pair of points. These objects arise naturally in many areas of mathematics. In Section 9.1 we prove a classical result due (essentially) to S. Banach that under appropriate hypotheses, an endomorphism ψ of a nonempty metric space has a unique fixed point x—indeed x is the limit (in the sense of ever decreasing distance) of the sequence x 0,ψx 0,ψ 2 x 0,… with any starting x 0. To illustrate the scope of this theorem we devote each of the remaining sections to an application. Section 9.2 establishes that an initial-value problem of ordinary differential equations has a unique solution. Section 9.3 introduces a tree-syntax for recursive specification in which repeated execution produces an infinite (but nonrecursive) syntax tree. The independence of the resulting infinite tree from “calling strategy” follows from the uniqueness assertion in the Banach theorem. Finally, we show in Section 9.4 that the language defined by a context-free grammar often arises as the unique fixed point of the Banach theorem.

We now generalize the results of Section 9 to consider the n-Dimensional (n > 1) Riemann-Stieltjes Integral with respect to an n-dimensional c.d.f., then, more generally, with respect to left-continuous n-dimensional b.v.f.’s. The development closely parallels that of the 1-dimensional case, and for this reason we will generally be briefer with proofs and descriptions than before. However, this by no means indicates that the n-dimensional Integral is any less important in applications.

We write a statistic as a functional on the sample paths of a stochastic process in order to break an analysis of the statistic into two parts: the study of continuity properties of the functional; the study of the stochastic process as a random element of a space of functions. The method has its greatest appeal when many different statistics can be written as functionals on the same process, or when the process has a form that suggests a simple approximation, as in the goodness-of-fit example from Chapter I. There we expressed various statistics as functionals on the empirical process U n , which defines a random element of D[0, 1]. Doob’s heuristic argument suggested that U n should behave like a brownian bridge, in some distributional sense.

As necessary preliminaries to a statement and proof of the inversion theorem, which together with its elementary properties makes the Laplace transform a powerful tool in applications, we must first take note of some results from classical analysis.1 Suppose that f(x) is a function continuous on the closed interval a ≤ x ≤ b (and hence uniformly continuous).