REAL ANALYSIS 1 - Spring 2010

Henri Lebesgue, 1875-1941

COURSE: MATH 5210-001

TIME: 9:45--11:05 TR, PLACE: Room 477 Brown Hall

INSTRUCTOR: Dr. Robert Gardner, OFFICE: Room 308F of Gilbreath Hall

OFFICE HOURS: T.B.A. PHONE: 439-6979 (Math Office 439-4349)

E-MAIL: gardnerr@etsu.edu
WEBPAGE: http://faculty.etsu.edu/gardnerr/gardner.htm

TEXT: Real Analysis, Third Edition, by H. L. Royden.

ABOUT THE COURSE: This class offers a standard introduction to the theory of functions of a real variable from the measure theoretic perspective. As commented on page 1 of the text, we will cover "a portion of the material that every graduate student in mathematics must know." Whereas the undergraduate real analysis class presents the results of calculus from a rigorous perspective, we will introduce fundamentally new ideas which are basic extensions of the results from calculus. In particular, we will put a weight or "measure" on certain sets of real numbers. This measure will be used to define a new type of integral called the Lebesgue integral. Recall that a function is Riemann integrable if and only if it is discontinuous on a "small" set (namely, a set of measure zero). The Lebesgue integral is much more flexible and will allow us to integrate a much larger class of functions. In addition, we will have a number of "convergence theorems" related to the Lebesgue integral, which are not true in the setting of Riemann integration.

GRADING: We will have two tests (T1 and T2) and homework (HW) will be taken up a regular intervals (weekly). Your average will be computed as follows:

AVERAGE = (T1 + T2 + 2HW)/4.
Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate (which means, based on how the university assigns grade points, 3 point intervals for plus and minus grades - for example, an A- corresponds to percentage grades of 90, 91, and 92).

A NOTE ABOUT HOMEWORK: While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. Several of the homework problems are fairly standard for this class, and you may find proofs online. However, the online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class.

TENTATIVE OUTLINE:
Chapter 1: Set Theory (Sections 4, 5, and 6).
algebra of sets, σ-algebras, axiom of choice, countable sets, rational numbers.
Axiom of Choice: More Axiom of Choice and the Banach-Tarski Paradox.
Chapter 2: The Real Number System (Sections 1,2 3, and 7).
axioms of the real numbers, completeness, natural numbers, rational numbers, extended real numbers, Borel sets.
Chapter 3: Lebesgue Measure.
outer measure, measurable sets, Lebesgue measure, nonmeasurable set, Banach-Tarski Paradox, measurable functions, characteristic functions, Littlewood's principles, Egoroff's Theorem, Lusin's Theorem.
Chapter 4: The Lebesgue Integral.
Riemann integral, step functions, simple functions, Lebesgue integral of a bounded function, Bounded Convergence Theorem, Fatou's Lemma, Monotone Convergence Theorem, Lebesgue Convergence Theorem, general Lebesgue Integral, Convergence in measure.

IMPORTANT DATES:

SUPPLEMENTS
  1. Axiom of Choice handout is online at: PDF and PostScript.
  2. Banach-Tarski Paradox handout is online at: PDF and PostScript.
  3. A handout is available which discusses the inner measure approach to countable additivity: Measure Theory. The last few pages are still a rough draft.
  4. A Study Guide for Test 1 is available in PDF.
  5. A Study Guide for "Test 2" is available in PDF.

OTHER RESOURCES. The following were mentioned in class:

  1. To access the Mathematical Reviews: Go to the Sherrod Library online catalog. Click the "Title" tab and enter "Mathematical Reviews." Select "MathSciNet [Electronic Resource]" and follow the links. You will be asked to enter your user ID and password (the same you use for your ETSU e-mail). You are then redirected to MathSciNet and can freely use it and even download PDF versions of some of the papers you find! Of course, this protocol will work for any electronic journal available through the Sherrod Library.

Homework
Section
Problems
Due Date
Points
1.4
1.4.19a
Thursday 1/21
3=3
2.7 and 3.1
2.53, 3.1, 3.3
Tuesday 2/9
4+3+3=10
3.2
3.2.8
Thursday 2/18
3=3
3.3
3.3.11, 3.3.12, 3.3.14b
Friday 3/5
2+3+4=9
3.4
3.4.15, 3.4.16, BONUS: 3.4.17b
Thursday 3/25
3+3+(3)=6+(3)
4.3
4.3.3, 4.3.5, 4.3.6
Thursday 4/8
3+3+3=9
4.4
4.4.10a, 4.4.12, 4.4.15a
Tuesday 4/20
3+3+3=9
Hilbert Space
Problem 2 and Problem 3
Thursday 5/6
3+3=6
TOTAL
-
-
55+(3)
The numbers in parentheses represent bonus problems.
Problem 1. Prove that the inner product is continuous. That is, if áunñ → u and ávnñ → v, then áun, vnñ → áu, vñ. (This will act as Problem 5 on the final and is the take home portion of the final.)
Problem 2. Show that the set { 1⁄√2π, cos x⁄√π, sin x⁄√π, cos 2x⁄√π, sin 2x⁄√π, ...} is an orthonormal set in L2([-π, π]).
Problem 3. Find the projection of f(x)=x2 onto sin x in L2([-π, π]).

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Last updated: April 28, 2010.