# Math 618

**Instructor:** Clayton Shonkwiler

**Time:** Monday, Wednesday, Friday, 3:00–4:00

**Location:** Engineering E206

**Office:** Weber 206C

**Office Hours:** Wednesday 1:00–2:00 and Thursday 9:30–10:30 in Weber 017, or by appointment

**Email Address:** clayton.shonkwiler@colostate.edu

**Text:** *Topics in Linear and Nonlinear Functional Analysis*, by Gerald Teschl

**Syllabus**

## Overview

The course will be an introduction to Hilbert and Banach spaces, with applications to Fourier analysis and partial differential equations.

We will start with the basic theory of infinite-dimensional vector spaces and operators and come to grips with standard function spaces like \(L^2\) and their connections to Fourier analysis. Then we will develop the theory of compact operators, with applications to Sturm–Liouville problems and linear PDEs. After introducing the main technical tools for Banach spaces, including Baire’s theorem and the Hahn–Banach theorem, we will end with some spectral theory, including the Gelfand representation theorem.

Familiarity with basic real analysis (e.g., baby Rudin) is important, as is good working knowledge of (finite-dimensional) linear algebra.

The following books may be useful additional resources:

*Topics in Real Analysis*, by Gerald Teschl*Principles of Functional Analysis*, by Martin Schechter*Functional Analysis, An Introduction*, by Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis*Lecture Notes on Functional Analysis*, by Alberto Bressan*An Introductory Course in Functional Analysis*, by Adam Bowers and Nigel J. Kalton