21-640: Introduction to Functional Analysis
MWF 2:30-3:20 pm in Hamburg Hall room 237 (moved back from PPB 300 )
- Office: 6130 Wean Hall
- Phone: (412)268-2553
- email: rpego AT cmu.edu
- Office hours: TWTh 3:30-4:30pm and by appointment
Assignment 1, due W 2/7   pdf file
Assignment 2, due W 2/28   pdf file   (typo in P4(a): fixed 2/16)
Assignment 3, due M 3/26   pdf file
Assignment 4, due F 4/27   pdf file
Assignment 5, due W 5/2   pdf file
The final exam is scheduled for Friday, May 11, 5:30-8:30pm, Porter Hall A18A.
A syllabus is available here: pdf
The midterm test will be held Friday, March 30 in class.
A syllabus is available here: pdf
Reference: P. R. Chernoff, A simple proof of Tychonoff's theorem via nets,
American Mathematical Monthly 99 (no. 10) (1992) pp. 932--934. (Available online via JSTOR)
Highly recommended reference works:
M. Reed and B. Simon,
Methods of Mathematical Physics I: Functional Analysis,
published by Academic Press.
Other texts to consider:
N. Dunford and J. T. Schwartz,
Linear Operators. Part I: General Theory,
Wiley Interscience. (Huge. Pretty essential.)
T. Kato, Perturbation Theory for Linear Operators,
Springer-Verlag. (Especially good on spectral theory,
applications to differential operators.)
H. Brezis, Analyse fonctionnelle, Theorie et applications,
Masson, Paris, 1983. (In French, but beautiful.)
P. Lax, Functional Analysis, Wiley Interscience, 2002.
(Interesting applications, viewpoint of a master mathematician.)
W. Rudin, Functional Analysis, McGraw-Hill, 1973.
A. W. Naylor and G. R. Sell, Linear Operator Theory in
Engineering and Science, Springer-Verlag, 1982.
(More basic, tilted slightly toward science.)
A. Friedman, Foundations of Modern Analysis,
Dover, 1982. (Just the bare bones, but cheap.)
Functional analysis is a large subject and this is only an introduction to a
central set of topics. Research has largely tapered off in the subject
itself, but it is widely used in many areas of application.
Linear spaces: Hilbert spaces, Banach spaces, Topological vector spaces
(esp. Schwartz' distributions).
Hilbert spaces: geometry, projections, Riesz representation theorem, bilinear and quadratic forms, orthonormal sets and Fourier series.
Banach spaces: continuity of linear mappings,
Hahn-Banach, uniform boundedness, open-mapping theorems.
Compact operators, unbounded operators, closed operators.
Dual spaces: weak and weak-star topologies, reflexivity, convexity.
Adjoints of operators: basic properties, null spaces and ranges. Sequences of bounded linear operators: weak, strong and uniform convergence.
Solvability criteria for linear equations:
spectrum and resolvent of bounded operators,
spectral theory of compact operators, Fredholm alternative.
Applications: important function spaces in analysis,
differential operators, calculus of variations, evolution equations.
Real analysis (21355), linear algebra (21341), metric-space topology (21465).
(Course numbers are only examples.)
Also useful, but not essential, would be some knowledge
of complex variables and/or Fourier series.
Grading:   Based on
- Homework (about 6 sets)
- Mid-term exam
- Final exam
PDF files will be posted online at   http://www.math.cmu.edu/~bobpego/21640/ .
There will be a mandatory comprehensive final examination;
time and date to be announced.
Make-up tests are only possible in the case of a documented
medical excuse, a university-sanctioned absence (e.g., participation
in a varsity sporting event), or a family emergency. Please contact me
at the earliest time possible to schedule a make-up.