Stephen P. Shipman
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Department of Mathematics
Louisiana State University
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Real Analysis I

Math 7350
Louisiana State University
Fall Semester, 2015

Prof. Stephen Shipman


Place: Lockett 113
Time: 10:30-11:50 Tuesday, Thursday

Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@math.lsu.edu
Office Hours: Tuesday and Friday 1:00-3:00, or by appointment


Textbook

Real Analysis (second edition) by Gerald B. Folland; Wiley.

Course Description

Abstract measure and integration theory with application to Lebesgue measure on the real line and Euclidean space.

Prerequisite

The prerequisite is a good undergraduate math education, including a good undergraduate analysis course.

Course Topics

  1. Chapter 0: Set theory; cardinality, axiom of choice, cardinality, etc.
  2. Chapter 1: Abstract measure spaces
  3. Chapter 2: Integration on measure spaces
  4. Chapter 3: Signed measures and differentiation
  5. Chapter 4: Some point-set topology
  6. Chapter 5: Some functional analysis
  7. Chapter 6: Lp spaces
  8. Chapter 7: Radon measures
  9. Chapter 11: s'more measures

Literature

There is a bibliography of relevant works below, with links to some PDF files of excerpts.

Assignments

The book has lots of exercises and problems; see the schedule of weekly problem sets below. A grader will assist with the evaluation of the assignments.

These are the rules.

  1. Students may discuss problems with each other and other people (including me, of course) and consult other literature; in fact students are encouraged to search the literature and discuss ideas.
  2. However, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding.
  3. Any collaboration with others or consultation with sources must be noted on the assignment.

Evaluation

Regular assignments: 60%
Final exam: 40%
Final exam time: Monday, December 7 from 10:00 to 12:00 in Lockett 113

Grading scale:
A+: at least 95% A: at least 90% A-: at least 88%
B+: at least 85% B: at least 80% B-: at least 78%
C+: at least 75% C: at least 70% C-: at least 68%
D+: at least 65% D: at least 60% D-: at least 50%
F: less than 50%

The final exam will be based on the following themes:

Integral convergence theorems. Know the proofs of the Monotone Convergence Theorem, Fatou's Lemma (using MCT), and the Dominated Convergence Theorem (using Fatou's Lemma). Be able to use these theorems in specific examples and to work with counterexamples when the hypotheses of the statements are not satisfied.

Signed measures on the real line. Study functions of bounded variation and signed measures on R, particularly the Lebesgue-Radon-Nikodym Theorem(s) and the Fundamental Theorem of Calculus for Lebesgue Integrals (sections 3.2 and 3.5).

Lp spaces for a measure space (X,M,μ). For example: let Σ be the space of simple functions that vanish off of a set of finite measure; prove that the completion of Σ in the norm ||.||p is (isomorphic to) Lp. Know the Riesz Representation Theorem for Lp with 1 <= p < ∞ and elements of its proof.

Problems to do

Assignments are due by 5:00 PM on the due date listed.

Solutions of some problems are here: Solutions.pdf

Due date Section Problems to do
Fri., Sept. 4 Chapter 0 ProblemSet1.pdf
Wed., Sept. 23 Chapter 1 Exercises 5, 8, 12, 18, 28, 30, 31, 33
none Chapter 1 Exercises 3, 17, 26, 27
Fri., Oct. 16 Chapter 2 Exercises 3, 4, 13, 14, 21, 28, 34, 44, 55
none Chapter 2 Exercises 15, 19, 20, 26, 31, 33, 36, 39, 40, 42
Mon., Nov. 9 Chapter 3 Exercises 4, 7, 11, 13, 17, 24, 29, 31, 33, 35, 41
none Chapter 3 Exercises 1, 6, 9, 10, 12, 14, 16, 25, 28, 30, 32, 39, 42
Wed., Dec. 2 Chapter 6 Exercises 2, 3, 9, 19, 21, 22, 36, 38, 6
none Chapter 6 Exercises 41, 20, 35, 39

Bibliography

  1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications (second ed.), John Wiley, 1999.
  2. Loren Graham and Jean-Michel Kantor, Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity , Harvard Univ. Press, 2009.
  3. Michael Reed and Barry Simon, Methods of Modern Mathematics: Vol. I Functional Analysis, Vol. II Fourier Analysis and Self-Adjointness, Vol. III Scattering Theory, Vol. IV Analysis of Operators , Academic Press, 1980.
  4. A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, §36: The Stieljes integral , 1968 (translation by Silverman, Dover 1975).
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