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.