Fall 2006:  202, Functional Analysis

Course: MAT-202-1, CRN 43695
Instructor: Roman Vershynin
    e-mail: vershynin at math dot ucdavis dot edu
    Office hours: M 1-2, W 2-3 in 2218 MSB
Meeting times: MWF 11:00 - 11:50.
Location: BAINER 1128

Textbook: Yuli Eidelman, Vitali Milman, Antonis Tsolomitis, Functional analysis. An introduction. Graduate Studies in Mathematics, 66. American Mathematical Society, Providence, RI, 2004. xvi+323 pp. ISBN 0-8218-3646-3

Monographs for further reading and references:

• P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, 25. Cambridge University Press, 1991. ISBN 0-521-35618-0
• J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Springer reprinted 1977, 1979 edition. ISBN 3-540-60628-9
  

Course description:

1. Fundamental theorems of functional analysis. Open mapping theorem. Closed graph theorem. Projections in Banach spaces. Banach-Steinhaus theorem. Applications to constructing counterexamples in Fourier analysis. Hahn-Banach theorem. Separation of convex sets. Alaoglu theorem. Eberlein-Smulian theorem. Extremal points and Krein-Milman theorem. [Chapter 9]
   
2. Compact operators (Review Section 4.3). Adjoint operators; Shauder's duality theorem (Section 4.4). Spectrum of linear operators on Banach spaces. Fredholm theory of compact operators [Chapter 5]
3. Functions of operators. Spectral theory of self-adjoint bounded operators on Hilbert space. [Chapter7]

Prerequisites: MAT 201 A,B

Assessment:
Problem Set 1 (50%),
Problem Set 2 (50%)
Homework from the textbook:

• October 6: Chapter 9, Exercises 1, 2, 3, 5, 7, 8, 13, 14, 15 
  
• October 13: Chapter 9, Exercises 17, 18, 20, 21
• October 22: Try to prove Lemma 1.4.1 without looking at its proof on p.14.
  Ex. 1.6.23, 1.6.24, 9.10.23, 9.10.25
  

Web: http://www.math.ucdavis.edu/~vershynin/teaching/2006-07/202/course.html