** Lectures:**
MWF 2:30-3:20 pm in Hamburg Hall room 237 (*moved* back from PPB 300 )

** Instructor: **
**Robert Pego **

- Office: 6130 Wean Hall
- Phone: (412)268-2553
- email: rpego AT cmu.edu
- Office hours: TWTh 3:30-4:30pm and by appointment

**Assignments:**

- 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)

** Main Text:**

- M. Reed and B. Simon, Methods of Mathematical Physics I: Functional Analysis, published by Academic Press.

- 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. (Elegant. Terse.)
- 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.)

** Course Description: **
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.

**Prerequisites:**
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

**Homework:**
PDF files will be posted online at ** http://www.math.cmu.edu/~bobpego/21640/ **.

**Final Exam**:
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.