Functional Analysis
(old course number 327A and B)

Math 362a (Fall 1998)
Math 362b (Spring 1999)

taught by Eric Schechter

3 hours a week, time and place To Be Arranged

Prerequisites. Although the graduate catalog does not list any official prerequisites for this course, I have some unofficial recommendations. Functional analysis combines algebra, real analysis, and topology, so it requires some background in all three of those subjects. I would recommend that you should have already taken at least one course for each of those three subjects. However, we won't use the prerequisites very heavily at first, so if you are only missing a little of the required background, you'll probably have time to pick it up along the way.

Brief overview of the subject. In calculus and other early math courses, the answer to a problem is typically a number or a finite-dimensional vector. In some more advanced parts of analysis (e.g., partial differential equations), the answer to a problem typically is a function. Thus it behooves us to study classes of functions. Many of these classes of functions can be viewed as infinite-dimensional vector spaces -- e.g., the vector space of all polynomials in one real variable with real coefficients, or the vector space of all continuous functions from [0,1] into the scalar field. In many cases, the vector space can be equipped with a natural topology that expresses relations (e.g., convergence theorems) between the elements of that space. In the most elementary cases, the topology is metrizable and is given by a norm, but some important function spaces are unnormable or even unmetrizable.

In Math 362 we'll use many techniques for studying the structure of these function spaces. Some of the main themes are

(In infinite-dimensional normed spaces, not every closed and bounded set is compact.)

Another basic technique is that of linear functionals -- i.e., continuous linear maps from the topological vector space into the scalar field. The collection of all such continuous linear maps plays a role somewhat similar to the role played by the coordinate projections in finite-dimensional vector analysis -- i.e., the way that we analyze a three-dimensional vector by studying its three components.

Linear functionals are also used to determine natural weak topologies on vector spaces. These topologies have more compact sets; that fact is useful in existence proofs. But weak topologies generally are not metrizable, so they are conceptually a bit more advanced; for this reason we postpone their study until much later in the course.

Textbook: I will be writing my own textbook, and distributing it in class. Its contents will follow the syllabus listed further down on this web page.

(Some students may wonder how this new book will differ from my previous book, Handbook of Analysis and its Foundations, published in 1996. HAF was a huge reference book, a sort of mini-encyclopedia, covering functional analysis and many other topics at a level of generality that would prepare the reader for many years of research. The new book will be more of a textbook -- more exercises, fewer pages, narrower focus, functional analysis and little else. It will be more elementary -- it will not attempt to go to a level of generality beyond that required for a first course. In HAF I attempted to "tell the whole story" on a wide range of topics that are used in different parts of analysis, but in the new book I will attempt to get quickly to the most basic parts of functional analysis.)

As a backup, I am also ordering a book to be available at the school bookstore. I have selected

Introductory Functional Analysis with Applications, by Erwin Kreyszig, 704 pages, published by Wiley, ISBN 0-471-50459-9, Paperback: $50.95.
This book will be listed as optional at the bookstore; you may find it helpful if you discover that your background is weak or if you are not comfortable with the style of the book I'll be writing. (If my plans don't work out as expected, and I fall behind in my writing, we might switch to using Kreyszig's book as a required text -- but I think that is unlikely.) I chose Kreyszig's book because it is fairly readable, elementary, and (compared to other books) inexpensive. It does not follow my syllabus very closely, but neither does any other presently available book; that's why I've chosen to write my own.

Tentative Syllabus

I may cover very briefly, or skip entirely, any topics that I feel the class knows well enough; this may be particularly true for chapters that are labelled as "review." Nevertheless, those chapters will be included in the book for convenient reference, since it is my experience that different students have different backgrounds. The first few chapters are already done; later parts of the book are still being written.
  1. Review from Set Theory: A Few Subtleties of Informal Logic. Sets and Functions. Relations and Orderings. Monotone Functions and Sequences. Cardinality. The Axiom of Choice and What it Really Means. Filters and Ultrafilters.
  2. Review from Algebra: Groups. Fields. Ordered Fields and the Reals. Archimedean Fields (optional). Linearity.
  3. Convexity: Convex Sets. Convex Functions. Arithmetic of Convexity. Hahn-Banach Theorems. Some Inequalities for Integration Theory.
  4. Review from Topology: (Semi)metrics. Topologies. Examples of Topologies. Metric Topologies. Convergent Sequences. Continuity. Completeness. Some Applications of Completeness (Baire Category; Contraction Mappings). Completions.
  5. Banach Spaces: Norms. Basic Examples of Norms. Sup Norms. Operator Norms. Generalization to F-norms. More About Completeness (Closed Graph; Open Mapping; Uniform Boundedness). The Dual Space and Hahn-Banach Theorems.
  6. Calculus in Banach Spaces: This chapter is still being written. I haven't decided yet whether it will include the Henstock integral. It will certainly include the Riemann-Stieltjes integral, since we need that for path integrals. The chapter will also include Frechet derivatives, taking particular note of the difference between real and complex derivatives. Also, probably some versions of the Fundamental Theorem of Calculus, Chain Rule, Implicit Function Theorem, and Newton's Method for iterative solutions. Absolute convergence, unconditional convergence, and conditional convergence of series. Uniform convergence of series. The most basic properties of analytic Banach-space-valued functions.
  7. Review of Measure theory (in Banach Spaces): I haven't yet decided whether to include this chapter at all, nor where in the course it would belong. If I do include it, a lot of the proofs might be omitted. It is assumed that the reader has already seen measure theory in the real line; this chapter might just state (without proof) which of the main results generalize to Banach spaces. This chapter may be useful as a way of leading into spectral theory, covered later.
  8. Hilbert Spaces: Tentative partial list of topics: Orthonormal bases. Examples, involving trigonometric series and/or Gram-Schmidt orthonormalized polynomials. Perhaps the Sturm-Liouville Theorem, stated without proof. All Hilbert spaces are isomorphic to l2 spaces. Clarkson's inequalities (stated without proof) as a generalization of the Parallelogram equation.
  9. More Review from Topology: Compactness. Nets (Generalized Sequences). Convergent Nets. Tychonov's Theorem.
  10. Topological vector spaces: Tentative partial list of topics: Locally convex spaces. Metrizability (in terms of seminorms or F-seminorms). Characterizations of finite-dimensional TVS's. Duals of TVS's. Hahn-Banach Theorems. Some fixed point theorems (proofs omitted).
  11. Weak topologies: Tentative partial list of topics: Schur's Theorem. Banach-Alaoglu Theorem. James's Theorem (stated without proof). Characterizations of weak compactness and of reflexivity.
  12. Spectral theory: I'll definitely cover some of this subject, but I haven't yet decided how much or in what fashion.
  13. Distribution theory: I haven't yet decided whether I'll cover any of this subject. If I do, I may include not only some of the classical linear theory (of L. Schwartz) but also some of the more recent nonlinear theory (Rosinger and Colombeau).

Web page last updated 9 July 1998.