MA 546: Functional Analysis, Spring 2018


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.




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.


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.