MAT 1000 / MAT 457 Real Analysis I, Fall 2012

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

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

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

Office hours Monday 3-4:30 and 5:35-6:30pm

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

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

• Measure Theory: abstract measures and σ-algebras, Monotone Class Theorem, outer measures and Caratheorodry's theorem, Borel sets and Lebesgue measure
• Integration: convergence theorems, Fubini's theorem, change of variables and polar coordinates in Rⁿ
• Lebesgue Differentiation: Hardy-Littlewood maximal function, density points, Radon-Nikodym theorem
• Functional Analysis: Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L^p -spaces
  Hölder and Minkowski inequalities

We will also consult other sources, including

• Eliott H. Lieb and Michael Loss, Analysis. AMS Graduate Texts in Mathematics, Vol 14 (either edition)
• Elias Stein and Rami Shakarchi, Real Analysis. Princeton Lectures in Analysis, Vol. 3
• A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis. Dover Publications
• H.L. Royden: Real Analysis, Macmillan, 1988.

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 a 5-page essay (due Wednesday December 19)

20% : midterm test (Wednesday November 7, 5-7pm; 2202 Sanford Fleming)

40% : final examination (Wednesday, December 12, 2-5pm in WI 1017)

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.

Schedule:

Week 1 (September 10-14)
Chapter 1 -- Measures (Lieb & Loss, Chapter 1)

M: Why do we need integration and measure theory?

W: σ-algebras

F: The Monotone Class Theorem

Assignment 1 (due September 19)
Comments and hints

Week 2 (September 17-21)
Chapter 1 -- Measures

M: The Borel σ-algebra. Measures

W: Measures and outer measures

F: Carathéodory's theorem

Assignment 2 (due September 26)
Comments and hints, Solution to Problem 2

Week 3 (September 24-28)
Chapter 2 -- Integration

M: Proof of Carathéodory's theorem

W: Some corrections. Premeasures and elementary families

F: Lebesgue measure on Rⁿ: construction and geometric properties

Assignment 3 (due October 3)

Week 4 (October 1-5)
Chapter 2 -- Integration

M: Measurable functions

W: Construction of the integral. Monotone Convergence

F: Simple functions and really simple functions

Assignment 4 (due October 10)
Amendments

Week 5 (October 8-12)
Chapter 2 -- Integration

M: Thanksgiving holiday

W: Fatou's Lemma and Dominated Convergence

F: The space L¹

Assignment 5 (due October 17)
Corrections and comments.

Week 6 (October 15-19)
Chapter 6 -- L^p-spaces (Lieb & Loss, Chapter 2)

M: Norms and unit balls. L^p -spaces

W: L^p (Rⁿ) for finite p: Smooth functions are dense, translation is continuous

F: Hölder's inequality; the simplest interpolation inequality

Assignment 6 (due October 24)
Corrections

Week 7 (October 22-26)
Chapter 2 -- Integration

M: Product measures and Fubini's theorem

W: Fubini's theorem, cont'd.

F: Change of variables in Rⁿ

Assignment 7 (due October 31)

Week 8 (October 29-November 2)
Chapter 2 -- Applications of measure and integration theory

M: Integration in polar cordinates

W: Volume and surface area of the unit ball in Rⁿ

F: Littlewood's Three Principles

(no assignment due on November 7)

Week 9 (November 5-9)
... nothing new ...

M: Infinite products

W: Midterm test 5-7pm (2202 Sanford Fleming)

F: No lecture

Choose essay topic (one-paragraph outline due on November 14)
Last year's midterm test

Week 10 (November 12-16)
Chapter 5 -- Functional Analysis

M: Fall Break

W: The Kolmogorov Extension Theorem (correctly, this time)

F: Excursion: The Baire Category Theorem (lecture by Daniel Soukup)

Assignment 8 (due November 21)
Corrections

Week 11 (November 19-23)
Chapter 3 -- Signed measures and differentiation

M: Signed measures and complex measures

W: The Lebesgue-Radon-Nikodym theorem

F: Differentiation in Euclidean space. The Hardy-Littlewood maximal function

Assignment 9 (due November 28)
Corrections and clarifications

Week 12 (November 26-30)
Chapter 3 -- Signed measures and differentiation

M: Lebesgue density

W: Excursion: Applications of the Baire Category Theorem (lecture by Daniel Soukup)

F: Functions of bounded variation

Assignment 10 (due Friday December 7, in Daniel's mailbox)
Correction

Week 13 (December 2-6)
Chapter 3 -- Signed measures and differentiation

M: The Fundamental Theorem of Calculus

W: Fundamental Theorem of Calculus, cont'd

Wednesday December 12

Final Exam 2-5pm (WI 1017)
(last year's exam, UVa problems)

Essays (due December 19, my box)
Office hours December 17/18: Mon 12-5pm, Tue 3-5pm.