Math 231B: Functional Analysis
Contents of this page
- Lecture time and location
- Office hours
- Class schedule
- Grading policy
- Academic integrity
- Anonymous feedback
- (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
- (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
- (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
You will find most information you would ever want to know about
LaTeX at the LaTeX project
page. Their page of guides is
- (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.
- 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
- 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
MW 11:45-12:45 and 2:45-4:00, and by
||Barbara D. MacCluer: Elementary Functional Analysis,
Springer, GTM vol. 253, 2009
- 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).
This schedule is only approximate.
||Introduction and 1.1 Normed linear spaces
|| 1.2 Orthogonality
||1.3 Hilbert space geometry
|| (Labor Day, no class)
||1.4 Linear functionals
||1.5 Orthonormal bases
||2.1 Bounded linear operators
||2.1 Adjoints of Hilbert space operators
||2.2 Adjoints of Banach space operators
||3.1 The Hahn-Banach theorem
||3.2 The principle of uniform boundedness
||3.3 Open mapping and Closed graph theorems
||3.4 Quotient spaces
||Discussion, problems, questions, etc.
||4.1 Finite-dimensional spaces and 4.2 Compact operators
||4.3 A preliminary spectral theorem
||4.4 The invariant subspace problem and 4.5 Introduction to the spectrum
||4.6 The Fredholm alternative (and applications)
||Discussion, problems, questions, etc.
||5.1 Banach and C* algebras: First examples
||5.2 Results on spectra
||5.3 Ideals and homomorphisms
||5.4 Commutative Banach algebras
||5.5. Weak topologies
||5.6 The Gelfand transform
||5.7 The continuous functional calculus
||5.8 Fredholm operators
||6.1 Normal operators as multiplication operators
||6.2 Spectral measures
||Functional analysis and quantum mechanics
||Discussion, problems, questions, etc./presentations
- Based on homework (50%), and a class paper and presentation
(50%). There will be no exams.
- None yet.
- There will be regular homework assignments, taken mostly from
||Exercises 1.2, 1.3, 1.7, and 1.8
||Ex. 1.13, 1.16, 1.20, 1.21, and 1.28
||Ex. 2.2, 2.6, 2.8, and 2.10
||Ex. 2.15, 2.16, 2.23, and 3.8
||Ex. 3.14, 3.17, 3.23, and 3.27
||Ex. 4.6, 4.9, 4.10(b), 4.16, and 4.19
||Ex. 5.11, 5.16, 5.25, 5.28.
- Each student is expected to write a short paper (in LaTeX) on a
topic related to functional analysis. Here are some potential
- Applications to quantum mechanics
- Applications of Fredholm's integral equations to potential theory
- Applications to the virating string problem
- Survey of fixed point theorems
- Introduction to von Neumann algebras
- Introduction to calculus of variations
- History of functional analysis
By default, I regard my students as honest individuals and expect them
to abide by the University policy on
If you have any comments or suggestions, please fill out this
anonymous feedback form.
Last modified: Wed Dec 15 15:56:41 PST 2010