MATH 6220: Applied Functional Analysis (Spring 2012)

Instructor: Lars Wahlbin

This will be a traditional first course in functional analysis covering very basic things that any student aspiring to work in applied mathematics will need to know. Functional analysis is where linear algebra and basic analysis meet infinite dimensional spaces, and topology comes into play, too. There is a rich and fascinating interplay between these, but we will keep in mind that the subject of functional analysis is rooted, founded, in applications. Historically, these applications were primarily in physics and engineering, but these days economics, finance, numerical snalysis, and operations research are there too.

We shall use the latter half of a book written by a master of applied mathematics:

Avner Friedman, Foundations of Modern Analysis, Dover (the price is right).

To whit, we will do Chapter 4 (Elements of Functional Analysis in Banach Spaces), Chapter 5 (Completely Continuous Operators), and Chapter 6 (Hilbert Spaces and Spectral Theory). I shall not assume that you know Chapters 1–3 in the book (i.e., Lebesgue measure and integration, and deeper properties of metric spaces). When necessary, at a very few instances, I will go back and describe what we need. The main point, used in examples of the general theory, is that some L_p spaces of Lebesgue integrable functions are complete — an isolated fact that is easy to apply.

Deviations from the text will be frequent and I hope interesting. In particular, I will expand on and do applications of Duality theory for conjugate and reflexive spaces, and the “weak” topology there. A main application is to a nonlinear partial differential equation.