## Fall 2008 Math 645

# What, When, and Where:

### Course Description:

Functional Analysis and Applications.

### Text:

### Lectures:

MWF 10:00-10:50 331 TMCB

### Office Hours:

MWF 9:00-9:50 306 TMCB

### Grading Scheme:

100% Homework Assignments

# Course Schedule: (subject to change)

Sep 03 Inner Product Spaces, Cauchy Schwarz Inequality

Sep 05 Hilbert Spaces, Norms

Sep 08 Orthogonality, Projections

Sep 10 Riesz Representation Theorem

Sep 12 Orthonormal Sets

Sep 15 Orthonormal Bases

Sep 17 Directed Sets, Nets, Convergence

Sep 19 Equivalent Properties of Orthonormal Bases

Sep 22 Isomorphisms, Fourier Transform, Direct Sums

Sep 24 Operators on Hilbert Spaces

Sep 26 Adjoint of an Operator I

Sep 29 Adjoint of an Operator II

Oct 01 Projections and Invariant Spaces

Oct 03 Compact Operators I

Oct 06 Compact Operators II

Oct 08 Compact Operators III

Oct 10 Self-Adjoint Operators

Oct 13 Sturm-Liouville Theory I

Oct 15 Sturm-Liouville Theory II

Oct 17 Compact Normal Operators

Oct 20 Introduction to Banach Spaces

Oct 22 Spaces of Continuous Functions, Bounded Operators

Oct 24 Riesz's Lemma and Locally Compact Spaces

Oct 27 Product and Quotent Spaces

Oct 29 Linear Functionals I

Oct 31 Linear Functionals II

Nov 03 Hahn-Banach Theorem I

Nov 05 Hahn-Banach Theorem II

Nov 07 Hahn-Banach Theorem III

Nov 10 More on Duals

Nov 12 Open Mapping and Closed Graph Theorems

Nov 14 Review of Baire Category

Nov 17 Principle of Uniform Boundedness

Nov 19 Locally Convex Spaces I

Nov 21 Locally Convex Spaces II

Nov 24 Locally Convex Spaces III

Dec 01 Metrizable and Normable LCS I

Dec 03 Metrizable and Normable LCS II

Dec 05 Distributions I

Dec 08 Distributions II

Dec 10 Distributions III

# Assignments:

Assignment #1 (Due Sept 12)

1.1 # 3,6,8

1.2 # 1,2,4-6

Assignment #2 (Due Sept 19)

1.3 # 2-6

1.4 # 10-14

Assignment #3 (Due Sept 26)

1.4 # 15-17,19-20

1.5 # 2-4,6,7

Assignment #4 (Due Oct 3)

2.1 # 4,6,8,10

2.2 # 2,6,7,12-14

Assignment #5 (Due Oct 17)

2.3 # 1,4,6,7

2.4 # 2-4,6,8,13

Assignment #6 (Due Oct 31)

2.5 # 3-5

2.6 # 1-6 (6 counts as two problems)

Assignment #7 (Due Nov 14)

3.1 # 1,8,9,13,16

3.2 # 1

3.4 # 1-4

Assignment #8 (Due Dec 19)

4.1 # 2,3,5,23