# MA 546: Functional Analysis, Spring 2018

## Announcements

## General Course Info

**Office** My office is 446 in the Mathematical Sciences building.

**Office hours** MWF 2:30-3:30, or by appointment

**Text** There is no required text. The main (recommended) text for the class is *Functional Analysis* by Peter D. Lax. Two secondary texts are *A Course in Functional Analysis* by John B. Conway and *Linear Topological Spaces* by J.L. Kelley and Isaac Namioka.

**Grading** Grades will be determined 50% on course participation and attendance and 50% on homework assignments.

## Notes

Link.

## Agenda

The following is a tentative outline of topics covered and is subject to change.

**Topic 1: Hilbert Spaces** --- Examples; Convex subsets; Riesz representation; Orthonormal bases; Radon--Nikodym theorem.

**Topic 2: Banach Spaces** --- Normed spaces; Completions; Examples; Quotients; Isometries; Mazur--Ulam theorem.

**Topic 3: Hahn--Banach theorem** --- Extension of linear functionals; Geometric version; Positive linear functionals and ordered vector spaces; Banach limits; Finitely additive measures.

**Topic 4:Dual Spaces** --- Bounded linear functionals; Reflexivity; Double Duals; Examples.

**Topic 5: Weak and Weak* Topologies** --- Weak convergence; Principle of Uniform Boundedness; Weak* convergence; Separability and Metrizability; Weak-closure of convex sets; Alaoglu's Theorem; Haar measure on a compact group; Eberlein--Smulian theorem.

**Topic 6: Locally Convex Topologies** --- Examples; Distributions; Krein--Milman theorem; Stone--Weierstrass theorem.

**Topic 7: Bounded Linear Maps** --- Principle of Uniform Boundedness; Open Mapping and Closed Graph theorems; Weak topologies; Compact Operators.

**Topic 8: Banach Algebras** --- Commutative Banach Algebras; Riesz functional calculus; Gelfand Spectrum; Convolution Algebras; Weiner's Tauberian theorem.

**Topic 9: Bounded Operators on Hilbert Space** --- Compact Operators; Hilbert-Schmidt operators; Normal operators; Weak topologies; Spectral Theorem; Spectral measure; Functional calculus; Applications.

**Topic 10: Unbounded Operators on Hilbert Space** --- Closeability; Cayley transform; Spectral theorem; Stone--von Neumann theorem; Applications.

## Homework

Complete as many of the assigned problems as you can. Collaboration is strongly encouraged. There are suggested due dates but no hard deadlines. However, assignments should be submitted in a timely manner.

Link to list of homework problems.