MATH 567: Introduction to Functional Analysis Fall 2021

Class schedule

TR 11:35–12:55, Burnside Hall 1B23

Date	Topics
R 9/2	Introduction. Weierstrass existence theorem.
T 9/7	Finite dimensional spaces. Riesz lemma. Uniform convexity.
R 9/9	Hilbert spaces. Projection. Riesz representation.
T 9/14	Sobolev spaces. Strong derivatives.
R 9/16	Mollifiers. Du Bois-Reymond lemma. Weak derivatives.
T 9/21	Meyers-Serrin theorem. Sobolev lemma. Regularity.
R 9/23	Neumann problems. Rellich lemma. Complex scalars.
T 9/28	Orthonormal bases.
R 9/30	Fourier bases.
T 10/5	Sturm-Liouville problems. Resolvent formalism.
R 10/7	Hilbert-Schmidt theory.
F 10/15	Gelfand triples. Singular Sturm-Liouville problems.
T 10/19	Functional calculus. Evolution equations.
R 10/21	Seminorms. Local spaces. Tempered distributions.
T 10/26	Hahn-Banach theorem.
R 10/28	Milman-Pettis theorem.
T 11/2	Weak convergence. Uniform boundedness principle.
R 11/4	Banach-Alaoglu theorem.
T 11/9	Tychonov's theorem. Consequences of the open mapping theorem.
R 11/11	Open mapping theorem. Bounded below property. Range closedness.
T 11/16	Closed range theorem.
R 11/18	Inf-sup conditions. Saddle point problems.
T 11/23	Perturbations of invertible operators. Fredholm alternative.
R 11/25	Compact operators.
T 11/30	Riesz-Schauder theory. Fredholm operators. Regularizers.
R 12/2	Perturbations of Fredholm operators. Bounded self-adjoint operators.

Recommended books

Barbara MacCluer, Elementary Functional Analysis. Springer.
Martin Schechter, Principles of Functional Analysis. AMS.
Peter Lax, Functional Analysis. Wiley.

Planned topics

Hahn-Banach theorem, duality
Weak and weak-star convergence
Bounded linear operators in Banach spaces
Compact operators, Fredholm theory
Spectral theory of bounded self-adjoint operators
Rudiments of locally convex spaces and unbounded operators
Semigroups (if time permits)

Course outline

Instructor: Dr. Gantumur Tsogtgerel

Prerequisite: MATH 455 (Honours Analysis 4) or equivalent

Calendar description: Banach and Hilbert spaces, theorems of Hahn-Banach and Banach-Steinhaus, open mapping theorem, closed graph theorem, Fredholm theory, spectral theorem for compact self-adjoint operators, spectral theorem for bounded self-adjoint operators.

Grading: Homework assignments 40% + Midterm Exam 20% + Course project 40%.

Homework: Assigned and graded roughly every other week, through MyCourses.

Midterm: Take-home type, in late Octorber or early November.

Course project: The course project consists of the student reading a paper or monograph on an advanced topic, typing up notes, and perhaps giving a lecture.