MAT 1001 / MAT 458 Real Analysis II, Spring 2013

How to reach me: Almut Burchard, 215 Huron Rm. 1024, 6-4174.

almut @math , www.math.utoronto.ca/almut/

Lectures MWF 12:10-1PM, BA 6183 .

Office hours Mondays 4-6pm

Teaching assistant: Daniel Soukup, daniel.soukup @mail.utoronto.ca .

Textbook:   G. Folland, Real Analysis: Modern Techniques and their Applications. Wiley (either edition).

• L^p -spaces: Minkowski inequality, Riesz representation theorem, complex interpolation, weak topology.
• Hilbert spaces: Adjoints, self-adjoint and unitary operators, orthogonal projections, closed subspaces and orthogonal complements, orthonormal bases, Bessel's inequality, the spectral theorem for compact operators.
• Fourier analysis: Fourier series and Fourier transform, Parseval's / Plancherel's identities and the Fourier inversion theorem, convergence results, L^p -theory, convolutions, some inequalities.
• Abstract functional analysis: Banach spaces, duals, weak topology, weak compactness, Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem, compact operators.
• Geometric analysis: Brunn-Minkowski inequality, isoperimetric inequality, symmetrization, sharp integral inequalities.
• Distributions: Test functions, weak derivatives, Schwarz space, Sobolev spaces and Sobolev inequalities.

We will also consult other sources, including

• Eliott H. Lieb and Michael Loss, Analysis. AMS Graduate Texts in Mathematics, Vol 14 (either edition) [Chapter 1, Chapter 2]
• Yitzhak Katznelson, An Introduction to Harmonic Analysis (either edition) Cambridge Press / Dover Publications
• Elias Stein and Rami Shakarchi, Real Analysis, Princeton Lectures in Analysis, Vol. 3 [Chapter 4]
• S. D. Promislow, A First Course in Functional Analysis
• H. L. Royden, Real Analysis (any edition), MacMillan

All of these are on reserve in the Mathematics library. I will post additional notes on the web, as needed.

Evaluation:

40% : Homework: weekly exercises (due Wednesdays in class), plus an oral presentation or short essay (your choice)

20% : midterm test

40% : final examination Thursday April 18, 7-10pm BA 2175

Remarks. Please discuss lectures and homework problems among yourselves and with me, and consult other sources. But write up your assignments 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.

Tentative Schedule:

Week 1 (January 7-11)
L^p -spaces (Folland Chapter 6, Lieb & Loss Chapter 2)

M: Minkowski's inequality

W: Hanner's inequality

F: Projection onto convex sets

Assignment 1 (due January 16)

Week 2 (January 14-18)
L^p -spaces, cont'd

M: Riesz representation theorem

W: Riesz representation theorem for L¹. Weak convergence

F: Mazur's theorem

Assignment 2 (due January 23)
Corrections

Week 3 (January 21-25)
L^p -spaces, cont'd

M: Alaoglu's theorem

W: Three-lines lemma and Riesz-Thorin interpolation theorem

F: Convolutions: Good kernels and approximate identities

Assignment 3 (due January 30)
Amendments

Week 4 (January 28-February 1)
Hilbert spaces (Folland Section 5.5, Stein & Shakarchi Chapter 4)

M: Inner products. Schwarz' inequality and parallelogram identity

W: Gram-Schmidt orthogonalization process, Bessel's inequality.

F: Completeness and Parseval's identity; Hamel bases vs. orthonormal bases.

Assignment 4 (due February 6)
Corrections

Week 5 (February 4-8)
Hilbert spaces, cont'd

M: Closed subspaces and orthogonal projections. Riesz reprentation theorem

W: Weak convergence. The adjoint of a linear transformation. Self-adjoint and unitary operators

F: The spectral theorem for compact self-adjoint operators

Assignment 5 (due February 13)
Comments and hints

Week 6 (February 11-15)
Fourier analysis (Folland Chapter 8, Stein and Shakarchi Section 3.3, Katznelson)

M: Proof of the spectral theorem

W: Fourier series

F: Fatou's theorem

Assignment 6 (due February 27)
Corrections

Winter break (February 17-21)

Week 7 (February 25-March 1)
Some abstract functional analysis (Folland Chapter 5, Promislow)

M: Hahn-Banach theorem

W: Banach-Alaoglu theorem

F: Open Mapping Theorem

Week 8 (March 4-8)
Excursion into geometric analysis

M:Brunn-Minkowski and isoperimetric inequality

W: Question hour (usual time & place)
Midterm test: 4-6pm BA 2155

F: No lecture

Choose presentation/essay topics (title due March 13)

Week 9 (March 11-15)
The Fourier transform (Lieb & Loss Chapter 5, Folland Chapter 8); Distributions (Folland Chapter 9, Lieb & Loss Chapter 6)

M: Definition and basic properties.

W: Construction of the L² -Fourier transform. Parseval's and Plancherel's identities

F: The Fourier inversion theorem

Assignment 7 (due March 20)

Week 10 (March 18-22)

M: The resolvent of the Laplacian. Hausdorff-Young inequality

W: Schwarz space and tempered distributions

F: Distributions and test functions Weak derivatives

Assignment 8 (due March 27)
Comments and corrections

Week 11 (March 25-29)
Distributions (Folland Chapter 9, Lieb & Loss Chapter 6)

M: Tempered distributions. The Fourier transform on S'.

W: Convolutions; approximation by smooth functions

F: Good Friday (no class)

Assignment 9 (due April 3)
Comments and hints

Week 12 (April 1-5)
Sobolev spaces (Lieb & Loss Chapters 4, 6, and 8)

M: Definition and basic properties. The space H^1

W: The Sobolev inequality for ||grad u||_2 (statement)

F: More about H^1.

Assignment 10 (CANCELED)

Final Exam: Thursday April 18, 7-10pm BA 2175

April 22-24
Student presentations / Essays due
( 10th floor lounge, 215 Huron, Guidelines)

Tuesday, April 9

1pm David Barmherzig, Halperin's lemma for the Spectral Theorem

Monday, April 22: (12-4)

Tuesday, April 23: (12-late)

12:00 Beatriz Lameda-Navarro, Kingman's Subadditve Ergodic Theorem

12:30 Fulgencio Lopez

13:00 Tracey Balehowsky, An Interpolation Proof of the Brunn-Minkowski inequality

13:30 Rosemonde Lareau-Dussault, Baire one functions

Wednesday, April 24: (10-2, 4-6)

11am Benjamin Schachter

11:30 Jeremy Lane, The direct method in the Calculus of Variations

12:o0

12:30 Quin Deng, The Kakeya problem and some applications to Fourier analysis

13:00

13:30 Andrew Colinet, Fixed points of the Fourier transform

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