MAT 1001 / MAT 458 Real Analysis II, Spring 2014

Almut Burchard, Instructor

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 Monday afternoons (TBA)

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

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 6-10)
A taste of abstract functional analysis

M: Metrics and norms; completeness

W: Baire Category Theorem

F: Open Mapping Theorem

Assignment 1 (due January 15)

Week 2 (January 13-17)
Hilbert spaces (Folland Section 5.5, Stein & Shakarchi Chapter 4)

M: Closed Graph Theorem

W: Inner products. Schwarz' inequality, parallelogram identity

F: Gram-Schmidt process. Hamel bases vs. orthonormal bases

Assignment 2 (due January 22)

Week 3 (January 20-24)
Hilbert spaces, cont'd

M: Closed subspaces and orthogonal projections. Riesz representation theorem

W: Bessel's inquality and Parseval's identity. Existence of orthonormal bases. Weak convergence.

F: The adjoint of a linear transformation.

Assignment 3 (due January 29)
Corrections

Week 4 (January 27-31)
Hilbert spaces, cont'd

M: Self-adjoint operators.

W: The spectral theorem for compact self-adjoint operators

F: Proof of the Spectral Theorem

Assignment 4 (due February 5)
For Problem 3bc, look at [Netyanun-Solmon 2006, p. 645 bottom].
Related results and open questions are discussed in [Bauschke-Matouskova-Reich 2004].

Week 5 (February 3-7)
More Hilbert spaces, and a litte abstract functional analysis (Lieb \& Loss Sections 2.13 and 2.18, Folland Section 5.2)

M: Weak convergence

W: Weak compactness and Mazur's theorem

F: The Hahn-Banach theorem

Assignment 5 (due February 12)

Week 6 (February 10-14)
L^p -spaces (Folland Chapter 6, Lieb & Loss Chapter 2)

M: The complex Hahn-Banach theorem, and some examples

W: Norms and unit balls. Hölder's inequality; the simplest interpolation inequality

F: L^p for finite p: Smooth functions are dense, translation is continuous

Assignment 6 (due February 26)
Comments

Week 7 (February 24-28)
L^p -spaces, cont'd (Lieb & Loss, Chapter 2)

M: Minkowski's inequality; Hanner's inequality

W: Projection onto convex sets

F: Riesz representation theorem

Week 8 (March 3-7)
L^p -spaces, cont'd (Folland Section 6.5)

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

W: Question hour (usual time & place)
Midterm (5-7pm in BA 6183)

F: Weak convergence. Banach-Alaoglu theorem

Week 9 (March 10-14)
The Fourier transform (Lieb & Loss Chapter 5, Folland Chapter 8)

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 19)

Week 10 (March 17-21)
Distributions (Folland Chapter 9, Lieb & Loss Chapter 6)

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 26)
Corrections

Week 11 (March 24-28)
Distributions (Folland Chapter 9, Lieb & Loss Chapter 6)

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

W: Convolutions; approximation by smooth functions

F: How to compute distributional derivatives

Assignment 9 (due April 2)

Week 12 (March 31 - April 4)
Sobolev spaces, and a little geometric analysis (Lieb & Loss Chapter 6; Stein & Shakarchi Section 1.5)

M: Definition and basic properties. Statement of the Sobolev inequality for ||grad u|| _p

W: Brunn-Minkowski and isoperimetric inequality

F: Proof of Brunn-Minkowski inequality

Assignment 10 (due April 9)
Final Exam: Wednesday, April 9, 2-5pm in Bahen 6183
( Last year's final exam, qualifying exams)

Presentations: April 16 (Wed. morning), April 17 (Thu. morning), April 21 (Mon. all day)
Instructions and schedule

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