Math 21-640: Functional Analysis

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. Topics covered include:

1. Linear spaces: Hilbert spaces, Banach spaces, Topological vector spaces (esp. Schwartz'

distributions).

2. Hilbert spaces: geometry, projections, Riesz representation theorem, bilinear and quadratic

forms, orthonormal sets and Fourier series.

3. Banach spaces: continuity of linear mappings, Hahn-Banach, uniform boundedness, open-

mapping theorems. Compact operators, unbounded operators, closed operators.

4. 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.

5. Solvability criteria for linear equations: spectrum and resolvent of bounded operators, spectral

theory of compact operators, Fredholm alternative.

6. Applications: important function spaces in analysis, differential operators, calculus of variations,

evolution equations.

Texts/ References:

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

Academic Press. (MAIN TEXT)

2. N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Wiley Interscience. (Huge.

Pretty essential.)

3. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag. (Especially good on spectral

theory, applications to differential operators.)

4. H. Brezis, Analyse fonctionnelle, Theorie et applications, Masson, Paris, 1983. (In French, but

beautiful.)

5. P. Lax, Functional Analysis, Wiley Interscience, 2002. (Interesting applications, viewpoint of a

master mathematician.)

6. W. Rudin, Functional Analysis, McGraw-Hill, 1973. (Elegant. Terse.)

7. A. W. Naylor and G. R. Sell, Linear Operator Theory in Engineering and Science, Springer-Verlag,

1982. (More basic, tilted slightly toward science.)

8. A. Friedman, Foundations of Modern Analysis, Dover, 1982. (Just the bare bones, but cheap.)