MAT 1001 / MAT 458 Real Analysis II, Spring 2015

How to reach me: Almut Burchard, Bahen Center Rm. 6234, 8-3318.

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

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

Office hours Mondays 4-6pm

Teaching assistant: John Enns, john.enns @mail.utoronto.ca .

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

• Hilbert spaces: Fourier series. Adjoints, self-adjoint and unitary operators, orthogonal projections, closed subspaces and orthogonal complements, orthonormal bases, Bessel's inequality, the spectral theorem for compact operators.
• Abstract functional analysis: Banach spaces, duals, weak topology, weak compactness, Hahn-Banach theorem, open mapping theorem, closed-graph theorem, uniform boundedness theorem.
• Distributions: Test functions, weak derivatives, Schwarz space, Sobolev spaces.
• Geometric analysis: Brunn-Minkowski inequality, isoperimetric inequality, symmetrization.
• Integral inequalities: Interpolation in L^p -spaces: Riesz-Thorin and Marcinkiewicz interpolation; Sobolev, Hardy-Littlewood-Sobolev, Young and Hausdorff-Young 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, Chapter 6]
• Haïm Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Universitext [Chapter 1]
• Elias Stein and Rami Shakarchi, Real Analysis, Princeton Lectures in Analysis, Vol. 3 [Chapter 4]
• Yitzhak Katznelson, An Introduction to Harmonic Analysis. Cambridge Press / Dover Publications (either edition)
• S. D. Promislow, A First Course in Functional Analysis
• H. L. Royden, Real Analysis (any edition), MacMillan

Most 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),

           presentation (April 16-17, see description and schedule)

20% : midterm test Wednesday March 4 (old tests: 2013, 2014, solutions: 2015)

40% : final examination Wednesday April 8 (old exams: 2013, 2014)

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 5-9)
Fourier series (Folland Section 8.4-8.5, Stein & Shakarchi Section 4.3)

M: Fourier coefficients. completeness and Schauder bases

W: The Poisson kernel; Abel summation

F: Convergence of Fourier series

Assignment 1 (due January 14)

Week 2 (January 12-16)
Hilbert spaces (Stein & Shakarchi Section 4.4)

M: The adjoint of a bounded linear operator

W: Self-adjoint and unitary operators, compact operators

F: The spectral theorem for compact self-adjoint operators

Assignment 2 (due January 21)
Corrections and remarks
For Problem 4bc, look at [Netyanun-Solmon 2006, p. 645 bottom].
Related results and open questions are discussed in [Bauschke-Matouskova-Reich 2004]

Week 3 (January 19-23)
Some abstract functional analysis (Folland Chapter 5, Brézis Chapter 1)

M: Excursion: Sturm-Liouville eigenvalue problems

W: Hahn-Banach theorem

F: Geometric implications of the Hahn-Banach theorem

Assignment 3 (due January 28)
Corrections

Week 4 (January 26-30)
Abstract functional analysis, cont'd (Brézis Chapter 1)

M: Convex sets and separating hyperplanes

W: Legendre transform

F: Fenchel-Moreau theorem

Assignment 4 (due February 4)
Corrections and remarks

Week 5 (February 2-6)
Abstract functional analysis, cont'd (Folland Chapter 5, Lieb & Loss Chapter 2)

M: Weak topology

W: Alaoglu's theorem, weak compactness

F: Banach-Alaoglu theorem, sequential weak compactness

Assignment 5 (due February 11)
Corrections

Week 6 (February 9-13)
More about L^p -spaces (Folland Chapter 6)

M: Baire category theorem

W: Open mapping theorem

F: Complex interpolation, Riesz-Thorin theorem

Assignment 6 (due February 25)
!! Major correction !!

Winter break (February 16-20)

Week 7 (February 23-27)
Distributions (Folland Chapter 9, Lieb & Loss Chapter 6)

M: Distributions and test functions. Weak derivatives

W: Tempered distributions. The Fourier transform on S'

F: Convolutions

Assignment 7 (due March 11)
Corrections

Week 8 (March 2-6)
Distributions, cont'd (Folland Chapter 9; Lieb & Loss Chapter 6)

M: Examples. Approximation by smooth functions

W: No lecture (question hour; usual time & place)
Midterm test, 3:30-6:30 pm, BA 6183

F: Sobolev spaces (definition)

Week 9 (March 9-13)
Some geometric analysis (Stein & Shakarchi Section 1.5)

M: Brunn-Minkowski inequality

W: Isoperimetric inequality

F: The uniform surface measure on the n- dimensional unit sphere

Assignment 8 (due March 18)

Week 10 (March 16-20)
Sobolev spaces (Lieb & Loss Chapters 6 and 3)

M: The spaces W^k,p

W: Density of smooth functions.

F: Statement of the Sobolev inequality and HLS inequalities

Assignment 9 (due March 25)
Addendum

Week 11 (March 23-27)
Integral inequalities (Lieb & Loss, Chapters 3 and 4; Folland Chapter 6.3)

M: Symmetric decreasing rearrangement.

W: The Riesz-Sobolev inequality. Steiner symmetrization

F: Conformal transformations

Assignment 10 (due April 1)

Week 12 (March 30-April 3)

M: Conformal invariance of HLS

W: Competing symmetris; proof of HLS

Final Exam: Wednesday April 8, 3:30-6:30pm, BA 6183

Final Presentations Thursday/Friday April 16/17 [schedule]

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