MAT 1000 / MAT 457 University of Toronto 2011-12
MAT 1001 / MAT 458 Real Analysis II, Spring 2015
Almut Burchard, Instructor
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 Lp -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 Lp -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 Wk,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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