Infinite Dimensional Analysis

In this chapter we present some useful odds and ends that should be a part of everyone’s mathematical tool kit, but which don’t conveniently fit anywhere else. Our presentation is informal and we do not prove many of our claims. We also use this chapter to standardize some terminology and notation. In particular, Section 1.7 introduces a number of kinds of binary relations.

We begin with a chapter on what is now known as general topology. Topology is the abstract study of convergence and approximation. We presume that you are familiar with the notion of convergence of a sequence of real numbers, and you may even be familiar with convergence in more general normed or metric spaces. Recall that a sequence {x _n } of real numbers converges to a real number x if and only if { ∣x _n -x∣ } converges to zero. That is, for every ε > 0, there is some n ₀ such that ∣x _n -x∣ < ε for all n ≥ n ₀. In metric spaces, the general notion of the distance between two points (given by the metric) plays the role of the absolute difference between real numbers, and the theory of convergence and approximation in metric spaces is not all that different from the theory of convergence and approximation for real numbers. For instance, a sequence {x _n } of points in a metric space converges to a point x if and only if the distance d(x _n, x) between x _n and x converges to zero as a sequence of real numbers. That is, if and only if for every ε > 0, there is an n ₀ such that d(x _n, x) < ε for all n ≥ n ₀. However, metric spaces are inadequate to describe approximation and convergence in more general settings. A very real example of this is given by the notion of pointwise convergence of real functions on the unit interval.

In Chapter 2 we introduced topological spaces to handle problems of convergence that metric spaces could not. Nevertheless, everyone would rather work with a metric space if they could. The reason is that the metric, a real-valued function, allows us to analyze these spaces using what we know about the real numbers. That is why they are so important in real analysis. We present here some of the more arcane results of the theory. A good source for some of this lesser known material is Kuratowski [149]. Many of these results are the work of Polish mathematicians in the 1920’s and 1930’s. For this reason, a complete separable metric space is called a Polish space.

One way to think of functional analysis is as the branch of mathematics that studies the extent to which the properties possessed by finite dimensional spaces generalize to infinite dimensional spaces. In the finite dimensional case there is only one natural linear topology. In that topology every linear functional is continuous, convex functions are continuous (at least on the interior of their domains), the convex hull of a compact set is compact, and nonempty disjoint closed convex sets can always be separated by hyperplanes. On an infinite dimensional vector space, there is generally more than one interesting topology, and the topological dual, the set of continuous linear functionals, depends on the topology. In infinite dimensional spaces convex functions are not always continuous, the convex hull of a compact set need not be compact, and nonempty disjoint closed convex sets cannot generally be separated by a hyperplane. However, with the right topology and perhaps some additional assumptions, each of these results has an appropriate infinite dimensional version.

This chapter studies some of the special properties of normed spaces. All finite dimensional spaces have a natural norm, the Euclidean norm. On a finite dimensional vector space, the Hausdorif linear topology the norm generates is unique (Theorem 4.61). The Euclidean norm makes ℝⁿ into a complete metric space. A normed space that is complete in the metric induced by its norm is called a Banach space. Here is an overview of some of the more salient results in this chapter.

Normed Riesz spaces are Riesz spaces equipped with lattice norms. Of course, completeness of the norm-induced metric is not automatic, but when imposed upon a lattice norm it precipitates several surprising consequences. For instance: positive operators between complete normed Riesz spaces are automatically continuous; not every Riesz space can become a complete normed Riesz space; and a Riesz space can admit at most one lattice norm under which it is complete. As suggested by these consequences, such spaces are important in their own right and have a special name. A norm complete Riesz space is called a Banach lattice. A Fréchet lattice is a completely metrizable locally solid Riesz space.

A set function is a real function defined on a collection of subsets of an underlying measurable space. In this chapter we consider set functions that have some of the properties ascribed to area. The main property is additivity. The area of two regions that do not overlap is the sum of their areas. A charge is any nonnegative set function that is additive in this sense. A measure is a charge that is countably additive. That is, the area of a sequence of disjoint regions is the infinite series of their areas. A probability measure is a measure that assigns measure one to the entire set. Charges and measures are intimately entwined with integration, which we take up in Chapter 9. But here we study them in their own right.

In modern mathematics the process of computing areas and volumes is called integration. The computation of areas of geometrical figures originated almost 2,500 years ago with the introduction by Greek mathematicians of the celebrated “method of exhaustion.” This method also introduced the modern concept of limit. In the method of exhaustion, a convex figure is approximated by inscribed (or circumscribed) polygons whose areas can be calculated—and then the number of vertexes of the inscribed polygons is increased until the convex region has been “exhausted.” That is, the area of the convex region is computed as the limit of the areas of the inscribed polygons. Archimedes (287–212 B.C.) used the method of exhaustion to calculate the area of a circle and the volume of a sphere, as well as the areas and volumes of several other geometrical figures and solids. The method of exhaustion is, in fact, at the heart of all modern integration techniques.

In this chapter, we introduce the classicalL _p -spaces and study their basic properties. For a measure space (X, Σ, µ) and 0 <p < ∞, the space L _p (µ) is the collection of all equivalence classes of measurable functionsf for which thep-norm

Unless otherwise indicated, in this chapter X is a metrizable topological space, and P (X) (or simply P) is the set of all probability measures on the Borel sets B of X. As usual, C _b (X) denotes the Banach lattice of all bounded continuous real functions on X. The reason we focus on probability measures is that every finite measure is the difference of measures each of which is a nonnegative multiple of a probability measure. That is, the probability measures span the space of all signed measures of bounded variation.

Among the most important and simplest normed and Banach spaces are the sequence spaces—vector subspaces of the space ℝ^ℕ of all real sequences. The sequence spaces can be thought of as the “building blocks” of Banach spaces and Banach lattices. Whether they are embedded in a Banach space or a Banach lattice reflect the topological and order structure of the space. In this chapter, we introduce the classical sequence spaces, φ, c ₀, c, ℓ ₁, ℓ∞, and ℓ _p (0 < p < ∞). We isolate each one of these sequence spaces and investigate their basic properties.

A correspondence is a set-valued function. That is, a correspondence associates to each point in one set a set of points in another set. As such, it can be viewed simply as a subset of the Cartesian product of the two sets. It may seem a bit silly to dedicate a chapter to such a topic, but correspondences arise naturally in many applications. For instance, the budget correspondence in economic theory associates the set of affordable consumption bundles to each price—income combination; the excess demand correspondence is a useful tool in studying economic equilibria; and the best-reply correspondence is the key to analyzing noncooperative games. The theory of “differential inclusions” deals with set-valued differential equations and plays an important role in control theory.

A Markov system is a stochastic system for which the state of the system at any time t depends only on the immediate past. Such processes are called Markov processes. In the language of conditional expectation of random variables, a Markov process is a family {X _t } of random variables (indexed by time) with the property that for any measurable f, any t, and any h > 0, E (f (X _t+h ) ∣ X _s , s ≤ t) = E (f (X _t+h ) ∣X _t ) . This defines a family of transition functions relating the distribution of the process at time t to the probability distribution of the process at time t + h. The process is stationary if such transition functions do not depend on t. Markov processes are generally considered to belong to the realm of probability theory, but some useful results can be derived by purely analytic methods. The main idea is to abstract from the random variables and look at the transition function as a mapping from states to probability measures on the set of states.

Ergodic theory can be described as the discipline that studies the long run average behavior of dynamical systems. There is a set S of possible states of the system, and the evolution of the system is usually modeled as a function T: S→ S. If the system is in state s at time t, then Ts is the state of the system at time t+ 1. The sequence {s, Ts, T ² s, ...} is called the orbit of the state s.