Math 231B: Functional Analysis
Fall 2010

Contents of this page

News
Description
Lecture time and location
Prerequisite
Office hours
Textbook
Syllabus
Class schedule
Grading policy
Handouts
Homework
Paper
Academic integrity
Anonymous feedback

News

(12/15) If you haven't emailed me your paper already, please try to do so by the end of this week. If I owe you any homework solutions, let me know. Otherwise: HAPPY HOLIDAYS AND ENJOY YOUR WINTER BREAK!

(11/27) The last homework (#7) has been posted. It is due on the last day of class, December 8 (not 6, as it previously said, by mistake). Btw, the last day of instruction this semester is December 9.

(11/22) As you may know, I'm teaching Math 213B: Introduction to Riemannian geometry next semester, but I would like to move the class to an earlier time also on MW (currently the class is scheduled for MW 16:30-17:45). If you are thinking of taking this class, please let me know of your time constraints as soon as you can. Two top candidates for a new time slot are 12 PM or 3 PM, but 10:30 or 1:30 also may end up fitting in my schedule.

(11/18) I just posted a LaTeX template for the expository paper called paper.tex. (It was actually written for Math 231A, but it'll do for 231B.) There you'll find a few examples of how to write math formulas, definitions, theorems, etc. in LaTeX. There is also a file called ref.bib, which contains all the references. You need to use BiBTeX to include the references into the paper, but this should be the last thing to worry about (first write the paper, then take care of the references). When the file paper.tex is compiled using LaTeX, the final product looks like this.

You will find most information you would ever want to know about LaTeX at the LaTeX project page. Their page of guides is especially helpful.

(11/9) Homework 6 has been posted and is due around November 24.

(10/19) I posted homework 5, due November 3, more or less.

(10/5) I just posted homework 4. It's due somewhere in the vicinity of October 20.

(9/21) Homework 3 has been assigned and is due (roughly) on October 4.

(9/7) I just posted homework 2, due roughly on Monday, September 20.

(8/27) Homework 1 has been posted and is due on September 8. Please let me know if you have any questions.

(8/24) Welcome! If you are thinking about potentially taking this class but are not sure about it, please talk to me as soon as you can.

Description

Functional analysis is in a sense the calculus of infinitely many variables. More accurately, it is a branch of analysis which studies infinite dimensional (topological or normed) vector spaces and maps between them. Functional analysis provides an ideal setting for the synthesis of analysis and algebra. Its historical roots are in the study of spaces of and transformations on functions, such as the Fourier transform, and various differential and integral operators. The need to study such transformations originated in physics, both classical and (especially!) quantum, as well as other branches of mathematics, such as the calculus of variations, differential equations, and so on. In this course we will focus on linear functional analysis, which studies linear maps.


Lecture time and location

MW 1:30-2:45 in MH 235

Prerequisite

Math 231A or instructor consent.

Note: To make the course more accessible, I will not assume familiarity with measure theory. I'm willing to accept anyone with a basic background in analysis and linear algebra, and a certain level of mathematical maturity. Physics and engineering students are welcome.


Office hours

MW 11:45-12:45 and 2:45-4:00, and by appointment

Textbook

Barbara D. MacCluer: Elementary Functional Analysis, Springer, GTM vol. 253, 2009


Syllabus

Green sheet
Banach and Hilbert spaces. Linear functionals. Orthonormal bases. Bounded linear operators. Adjoints of Hilbert and Banach space operators. The Big Three: the Hahn-Banach Theorem, the Principle of Uniform Boundedness, the Open Mapping and Closed Graph Theorems. Quotient spaces. Compact operators: the spectral theorem for compact, self-adjoint operators, the invariant subspace problem, the Fredholm alternative. Banach and C* algebras: ideals and homomorphisms, the Gelfand-Neimark and Gelfand-Mazur theorems, spectral radius, the Gelfand transform, the continuous functional calculus, Fredholm operators. The spectral theorem for normal operators. Applications of functional analysis in quantum mechanics, differential and integral equations, etc. (time permitting).

Class schedule

This schedule is only approximate.

# Date Reading
1 8/25 Introduction and 1.1 Normed linear spaces
2 8/30 1.2 Orthogonality
3 9/1 1.3 Hilbert space geometry
4 9/6 (Labor Day, no class)
5 9/8 1.4 Linear functionals
6 9/13 1.5 Orthonormal bases
7 9/15 2.1 Bounded linear operators
8 9/20 2.1 Adjoints of Hilbert space operators
9 9/22 2.2 Adjoints of Banach space operators
10 9/27 3.1 The Hahn-Banach theorem
11 9/29 3.2 The principle of uniform boundedness
12 10/4 3.3 Open mapping and Closed graph theorems
13 10/6 3.4 Quotient spaces
14 10/11 Discussion, problems, questions, etc.
15 10/13 4.1 Finite-dimensional spaces and 4.2 Compact operators
16 10/18 4.3 A preliminary spectral theorem
17 10/20 4.4 The invariant subspace problem and 4.5 Introduction to the spectrum
18 10/25 4.6 The Fredholm alternative (and applications)
19 10/27 Discussion, problems, questions, etc.
20 11/1 5.1 Banach and C* algebras: First examples
21 11/3 5.2 Results on spectra
22 11/8 5.3 Ideals and homomorphisms
23 11/10 5.4 Commutative Banach algebras
24 11/15 5.5. Weak topologies
25 11/17 5.6 The Gelfand transform
26 11/22 5.7 The continuous functional calculus
27 11/24 5.8 Fredholm operators
28 11/29 6.1 Normal operators as multiplication operators
29 12/1 6.2 Spectral measures
30 12/6 Functional analysis and quantum mechanics
31 12/8 Discussion, problems, questions, etc./presentations


Grading policy

Based on homework (50%), and a class paper and presentation (50%). There will be no exams.


Handouts

None yet.


Homework

There will be regular homework assignments, taken mostly from the textbook.

# Due date Assignment
1 9/8 Exercises 1.2, 1.3, 1.7, and 1.8
2 9/20 Ex. 1.13, 1.16, 1.20, 1.21, and 1.28
3 10/4 Ex. 2.2, 2.6, 2.8, and 2.10
4 10/20 Ex. 2.15, 2.16, 2.23, and 3.8
5 11/3 Ex. 3.14, 3.17, 3.23, and 3.27
6 11/24 Ex. 4.6, 4.9, 4.10(b), 4.16, and 4.19
7 12/8 Ex. 5.11, 5.16, 5.25, 5.28.


Paper

Each student is expected to write a short paper (in LaTeX) on a topic related to functional analysis. Here are some potential topics:


Academic integrity

By default, I regard my students as honest individuals and expect them to abide by the University policy on academic integrity.


Anonymous feedback

If you have any comments or suggestions, please fill out this anonymous feedback form.
Slobodan N. Simić

Last modified: Wed Dec 15 15:56:41 PST 2010