# Functional Analysis course

Functional analysis deals with various infinite dimensional vector spaces, often spaces of functions, but sometimes other types, like spaces of measures, or distributions.

Here is the first main text.
Outline of functional analysis

The following provides some useful background in abstract analysis.
Metric spaces and compactness

Here is complementary material, on compact spaces that are not metrizable.
Compact Hausdorff spaces

Particularly important classes of spaces studied in functional analysis are Banach spaces and Hilbert spaces. Examples include Lp spaces. See the following files for background on Lp spaces.
Integration on measure spaces

Lp spaces I

Caratheodory construction of measures

Signed measures

Lp spaces II

Here is an important class of integrable functions
Uniformly integrable families of functions

Here is a variant of Lebesgue’s characterization of Riemann integrable functions, in the setting of a compact metric space.
Riemann integrable functions on a compact measured metric space: extended theorems of Lebesgue and Darboux.

Distributions are important objects that are more general than functions of class Lp. The theory of tempered distributions leads to a particularly beautiful approach to Fourier Analysis. The following material deals with Fourier analysis and distribution theory.
Fourier analysis, distribution theory, and constant coefficient linear PDE

A more elementary introduction to Fourier analysis is given in Sections 13-14 of my

Introduction to Complex Analysis

Here is a review I wrote in March, 2012, on a book on distribution theory.
Review of “Distributions – Theory and Applications.”

For comparison, here is a review I wrote on PDE books by Lars Hormander, back in 1985.

The following material deals with spectral theory.
Spectral theory
Section 1 of “Spectral theory” gives a proof of the spectral theorem for self adjoint operators (and unitary operators) on a Hilbert space. This proof makes essential use of Fourier analysis on tempered distributions, as developed in the chapter on Fourier analysis mentioned above.

The following notes also establish the spectral theorem for bounded self-adjoint and unitary operators, using more elementary techniques than the harmonic analysis techniques in Section 1 of “Spectral theory.”
The spectral theorem for self-adjoint and unitary operators

Another approach to the proof of the spectral theorem (for commuting families of self-adjoint operators), using Banach algebra techniques, is given in the following.
Banach algebras – bare bones basics

The following notes have material on harmonic analysis on groups.
Lectures on Lie groups

Functional analysis is valuable in the study of Ergodic Theory. Here is some material on this. The reference to exercises in Chapter 9 in the proof of Lemma 14.1 is to the notes “Lp Spaces II,” listed above.
Ergodic theory

Here are some notes on integration of Banach space valued functions.
Vector valued integration

Vector integration and disintegration of measures

Fredholm operators play an important role in applications of Functional Analysis. Often, analytic families of Fredholm operators arise. These notes deal with the multivariable case.
Multidimensional analytic Fredholm theory

Measures are a special case of a more general class of set functions called capacities. Various capacities that are not measures play an important role in potential theory. These notes introduce the notion of capacity.
Capacities