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

    Planned topics

    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.