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Fall 2008 Math 645

What, When, and Where:

Course Description:

Functional Analysis and Applications.

Text:

Cover of "Graduate Texts in Mathematics: A Course in Functional Analysis, Second Edition" by John B. Conway

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