MAT 1001 / MAT 458 University of Toronto 2022

MAT 1001 / MAT 458 Real Analysis II, Winter 2022

Almut Burchard, Instructor

To sit-in on zoom lectures without registering: Please email me for the zoom code. (Alternatively, if you have a UofT email address, I can add you to the class roster on Quercus).
How to reach me: Almut Burchard, Bahen Center Rm. 6234, 8-3318.
almut @math,
Office hours Mondays 5-6:30 (Passcode: 484448)
Lectures W 9:10-11am, F 9:10-10am
Online until February 18; resuming in-person on March 2 (ES B142). Zoom registration codes have been posted on Quercus
Teaching assistant: Tomas Dominguez y Chiozza, tomas.dominguezchiozza .
Textbook:   G. Folland, Real Analysis: Modern Techniques and their Applications. Wiley (either edition). We will focus on Chapters 5, 6, 8, 9.
Other sources include
Elliott H. Lieb and Michael Loss, Analysis. AMS Graduate Texts in Mathematics, Vol 14 (either edition)
Haïm Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Universitext
Elias Stein and Rami Shakarchi, Real Analysis. Princeton Lectures in Analysis, Vol. 3
H. L. Royden, Real Analysis. MacMillan (any edition)
(all on reserve in the Mathematics library). Additional notes will be posted, as needed
  1. Lp -spaces: Minkowski inequality, Hanner's inequality, Riesz representation theorem, interpolation theorems
  2. Abstract functional analysis: Banach spaces and their duals. The Hahn-Banach theorem and its geometric implications, Legendre-Fenchel transform. Baire Category theorem, open mapping theorem, closed-graph theorem, uniform boundedness theorem. Weak topologies, Banach-Alaoglu theorem.
  3. Hilbert spaces: Self-adjoint and unitary operators, orthogonal projections, Bessel's inequality, orthonormal bases. The spectral theorem for compact self-adjoint operators
  4. Fourier integral transform: L2 theory, Parseval's identity, inversion formula, convolutions and derivatives, Poisson summation. The Sobolev spaces Hs
  5. Distributions: Test functions, weak derivatives, convolutions, approximation by smooth functions. Schwartz space and tempered distributions, the Sobolev spaces Wk,p
  6. Geometric and functional inequalities on Rd: Young, Hausdorff-Young, Brunn-Minkowski, isoperimetric, Hardy-Littlewood-Sobolev, Sobolev, Poincaré. Compact embeddings, Rellich lemma
Attendance is expected. If you can't be at a lecture, please read up on the material and use my office hours for questions.
40% : weekly assignments (due Tuesdays 11:59pm on crowdmark; drop 1)
20% : midterm test Wednesday March 9 (during lecture time)
40% : final examination (April 11, 2-5pm; in-person if possible; room TBD; comprehensive)
Please discuss lectures and homework problems freely, and consult available sources. Then write up your solutions individually, in your own words, and be ready to defend them! Your work will be judged on the clarity of your presentation as well as correctness and completeness.

Late submissions will be marked summarily at the end of the term. Note that crowdmark stops accepting uploads at some point after the deadline. In that case, please email me your assignment, making sure each problem appears in a separate attachment. On the first page of the first problem, please state the reason for the delay, then date and sign.

Absences. If you miss a substantial part of the course, please keep me updated. A Verification of Illness or Injury form is currently not required, see the University's policy.
Academic integrity statement. Consult the website Student Academic Integrity for information.
Accessibility. If you require accommodations for a disability, or have any accessibility concerns about the course, the classroom or course materials, please contact Accessibility Services as soon as possible:, or