Beginning Functional Analysis

Functional analysis was developed in the last years of the nineteenth century and during the first few decades of the twentieth century. Its development was, in large part, in response to questions arising in the study of differential and integral equations. These equations were of great interest at the time because of the vast effort by many individuals to understand physical phenomena.

The goal of this chapter is to introduce the abstract theory of the spaces that are important in functional analysis and to provide examples of such spaces. These will serve as our examples throughout the rest of the text, and the spaces introduced in the second section of this chapter will be studied in great detail. The abstract spaces—metric spaces, normed spaces, and inner product spaces—are all examples of what are more generally called “topological spaces.” These spaces have been given in order of increasing structure. That is, every inner product space is a normed space, and in turn, every normed space is a metric space. It is “easiest,” then, to be a metric space, but because of the added structure, it is “easiest” to work with inner product spaces.

Let (M, d) be a metric space. Recall that the r-ball centered at x is the set for any choice of x ∈ M and r > 0. These sets are most often called open balls, open disks, or open neighborhoods, and they are denoted by the above or by B(x, r), D _r (x), D(x, r), N _r (x), N(x, r), among other notations. A point x ∈ M is a limit point of a set E ⊆ M if every open ball B _r (x) contains a point y ≠ x, y ∈ E. If x ∈ E and x is not a limit point of E, then x is an isolated point of E. E is closed if every limit point of E is in E. A point x is an interior point of E if there exists an r > 0 such that B _r (x) ⊆ E. E is open if every point of E is an interior point. A collection of sets is called a cover of E if E is contained in the union of the sets in the collection. If each set in a cover of E is open, the cover is called an open cover of 2s. If the union of the sets in a subcollection of the cover still contains E, the subcollection is referred to as a subcover for E. E is compact if every open cover of E contains a finite subcover. E is sequentially compact if every sequence of E contains a convergent subsequence. E is dense in M if every point of M is a limit point of E. The closure of E, denoted by Ē, is E together with its limit points. The interior of E, denoted by E° or int (E), is the set of interior points of E. E is bounded if for each x ∈ E, there exists r > 0 such that E ⊆ B _r (x).

The foundations of integration theory date to the classical Greek period. The most notable contribution from that time is the “method of exhaustion” due to Eudoxos (ca. 408–355 B.C.E.; Asia Minor, now Turkey). Over two thousand years later, Augustin Cauchy stressed the importance of defining an integral as a limit of sums. One’s first encounter with a theory of the integral is usually with a variation on Cauchy’s definition given by Georg Friedrich Bernhard Riemann (1826–1866; Hanover, now Germany). Though the Riemann integral is attractive for many reasons and is an appropriate integral to learn first, it does have deficiencies. For one, the class of Riemann integrable functions is too small for many purposes. Henri Lebesgue gave, around 1900, another approach to integration. In addition to the integrals of Riemann and Lebesgue, there are yet other integrals, and debate is alive about which one is the best. Arguably, there is no one best integral. Different integrals work for different types of problems. It can be said, however, that Lebesgue’s ideas have been extremely successful, and that the Lebesgue integrable functions are the “right” ones for many functional analysts and probabilists. It is no coincidence that the rapid development of functional analysis coincided with the emergence of Lebesgue’s work.

In the last section of Chapter 3 we introduced the Lebesgue L ^p -spaces for general measures and discussed their most basic properties. The most important L ^p -space, by far, is L ². Its importance is its role in applications, especially in Fourier analysis. The material of this chapter lies at the foundation of the branch of mathematics called harmonic analysis.

In this chapter you will read about the beginning material of operator theory. The chapter is written with the aim of getting to spectral theory as quickly as possible. Matrices are examples of linear operators. They transform one linear space into another and do so linearly. “Spectral values” are the infinite-dimensional analogues of eigenvalues in the finite-dimensional situation. Spectral values can be used to decompose operators, in much the same way that eigenvalues can be used to decompose matrices. You will see an example of this sort of decomposition in the last section of this chapter, where we prove the spectral theorem for compact Hermitian operators. One of the most important open problems in operator theory at the start of the twenty-first century is the “invariant subspace problem.” In the penultimate section of this chapter we give a description of this problem and discuss some partial solutions to it. We also let the invariant subspace problem serve as our motivation for learning a bit about operators on Hilbert space. The material found at the end of Section 3 (from Theorem 5.7 onwards) through the last section (Section 5) of the chapter is not usually covered in an undergraduate course. This material is sophisticated, and will probably seem more difficult than other topics we cover.

In this chapter we present a smorgasbord of treats. The sections of this chapter are, for the most part, independent of each other (the exception is that the third section makes use of the main theorem of the second section). The sections are not uniform in length or level of difficulty. They may be added as topics for lectures, or used as sources of student projects.