MAT 202: Functional Analysis (Winter
2004)

Course: MAT-202-1

Title: Functional Analysis

Quarter: Winter 2004

CRN: 52511

Instructor: Roman Vershynin (vershynin@math.ucdavis.edu)

Location: Wellman 127

Times: MWF 12:10 - 01:00 pm

Office hours: 671 Kerr Hall, MW 13:20 - 14:20 or by appointment

COURSE DESCRIPTION:

This is a basic course in functional analysis, which is aimed at the graduate students in all areas of Mathematics. The course will cover in particular:

- Linear spaces (Zorn Lemma, Hamel bases, linear operators, quotient spaces, hyperplanes)

- Normed and Banach spaces (norm and the unit ball, isometries, subspaces, quotients)

- Bounded linear operators (continuity, norm, extensions)

- Duality (Hahn-Banach theorem and its consequences, adjoint operators, subspaces versus quotients)

- Three fundamental principles (open mapping theorem, closed graph theorem, principle of uniform boundedness).

- Various consequences (such as Schauder bases, more on Fourier series, (un-)complemented subspaces)

- Compactness (in concrete spaces; compact operators, spectrum, Fredholm theory as time permits)

- Krein-Milman theorem, Banach-Alaoglu theorem (as time permits)

TEXTS (Optional):

1. John Conway, A course in Functional Analysis, Second edition, Springer-Verlag, 1990 (comprehensive, covers most of this course, although proofs and notation will be altered)

2. Walter Rudin, Functional Analysis, Second edition, McGraw-Hill, Inc., New York, 1991 (a classical text, profound, abstract, comprehensive, difficult)

PREREQUISITES: MAT 201 or consent of the instrucror

WEB: http://www.math.ucdavis.edu/~vershynin/teaching/202/course.html

ASSESSMENT:

Homework 40% (current assigment - see below)

Final 60%

Homeworks will be given out by-weekly on Fridays and will be due two weeks later at the start of the Friday class.

The final will be a take home exam, it will be due exactly one week after it is given out.

You can cooperate on the homeworks. No cooperation is allowed on the final.

Homework 1: due 01/30/2004

Homework 2: due 02/13/2004

Homework 3: due 02/27/2004

Homework 4: due 03/12/2004

Final Exam: due 03/23/2004

MWF 12:10 - 01:00 pm, Wellman 127

Course: MAT-202-1

Title: Functional Analysis

Quarter: Winter 2004

CRN: 52511

Instructor: Roman Vershynin (vershynin@math.ucdavis.edu)

Location: Wellman 127

Times: MWF 12:10 - 01:00 pm

Office hours: 671 Kerr Hall, MW 13:20 - 14:20 or by appointment

COURSE DESCRIPTION:

This is a basic course in functional analysis, which is aimed at the graduate students in all areas of Mathematics. The course will cover in particular:

- Linear spaces (Zorn Lemma, Hamel bases, linear operators, quotient spaces, hyperplanes)

- Normed and Banach spaces (norm and the unit ball, isometries, subspaces, quotients)

- Bounded linear operators (continuity, norm, extensions)

- Duality (Hahn-Banach theorem and its consequences, adjoint operators, subspaces versus quotients)

- Three fundamental principles (open mapping theorem, closed graph theorem, principle of uniform boundedness).

- Various consequences (such as Schauder bases, more on Fourier series, (un-)complemented subspaces)

- Compactness (in concrete spaces; compact operators, spectrum, Fredholm theory as time permits)

- Krein-Milman theorem, Banach-Alaoglu theorem (as time permits)

TEXTS (Optional):

1. John Conway, A course in Functional Analysis, Second edition, Springer-Verlag, 1990 (comprehensive, covers most of this course, although proofs and notation will be altered)

2. Walter Rudin, Functional Analysis, Second edition, McGraw-Hill, Inc., New York, 1991 (a classical text, profound, abstract, comprehensive, difficult)

PREREQUISITES: MAT 201 or consent of the instrucror

WEB: http://www.math.ucdavis.edu/~vershynin/teaching/202/course.html

ASSESSMENT:

Homework 40% (current assigment - see below)

Final 60%

Homeworks will be given out by-weekly on Fridays and will be due two weeks later at the start of the Friday class.

The final will be a take home exam, it will be due exactly one week after it is given out.

You can cooperate on the homeworks. No cooperation is allowed on the final.

Homework 1: due 01/30/2004

Homework 2: due 02/13/2004

Homework 3: due 02/27/2004

Homework 4: due 03/12/2004

Final Exam: due 03/23/2004