Real Analysis 2 - Spring 2001

COURSE: MATH 5220-001, CALL # 12104

TIME: 9:45--11:05 TR, PLACE: Room 313 of Gilbreath Hall

INSTRUCTOR: Dr. Robert Gardner

OFFICES: Room 308G of Gilbreath Hall and Room 201 of Brown Hall

PHONE: 439-6977 (308G), 439-8684 (201), Math Office 439-4349

OFFICE HOURS: 10:15-11:15 MWF, E-MAIL: gardnerr@etsu.edu

WEBPAGE: http://www.etsu.edu/math/gardner/gardner.htm

TEXT: Real Analysis, Third Edition, by H. L. Royden, Prentice Hall, 1988.

ABOUT THE COURSE: This term, we concentrate on function spaces. They can be similar to vector spaces, but the elements of the space are functions instead of vectors. We will cover the classical Banach spaces (Chapter 6), Hilbert spaces (we will use some supplemental material for this), product measures (Chapters 11 and 12), and other topics which we find of interest. I would recommend metric spaces (Chapter 7), topological spaces (Chapter 8), and compact spaces and manifolds (Chapter 9). Personally, I would also like to cover some applications of this material (such as applications of Hilbert space theory to quantum mechanics or the use of manifolds in differential geometry).

GRADING: Your grade will be based on your performance on the assigned homework.

TENTATIVE OUTLINE:

  • Chapter 6. The Classical Banach Spaces: the Lp spaces, Minkowski and Holder inequalities, completeness, bounded linear functionals on Lp.
  • Chapters 11 and 12. Measure, Product Measure, and Integration (partial): Radon-Nikodym Theorem, Lebesgue-Stieltjes integrals, product measures.
  • Hilbert Spaces. Inner product spaces, linear functionals, self-adjoint operators, eigenvalues, spectral decomposition.
  • Chapter 7. Metric Spaces: convergence, completeness, compact metric spaces.
  • Chapter 8. Topological Spaces: bases, separation axioms, connectedness.
  • Chapter 9. Compact and Locally Compact Spaces: Compact spaces, paracompact spaces, manifolds.
  • Chapter 10. Banach Spaces (maybe only partially covered): linear operators and functionals, Hahn-Banach Theorem, Hilbert spaces.

UPDATE!
We will spend some time studying functional analysis from Introduction to Hilbert Spaces with Applications by Lokenath Debnath and Priotr Mikusinski, published by Academic Press:
We will cover (as time permits):
  • Hilbert Spaces and Orthonormal Systems.
  • Linear Operators on Hilbert Spaces.
  • Mathematical Foundations of Quantum Mechanics (very briefly!).

Homework Assignments
Section
Problems
Due Date
5.4
12ab, 16a, 18, 20ab (choose 3)
Friday, January 19
6.1
1, 3, L0 "norm"
Friday, January 26
6.2
15b, 7b, Bonus: 8a, 8b
Friday, February 2
6.3
13, 14, Bonus: 11
Friday, February 9
6.5
22, 24
Friday, February 16
8.1
3, 7abc
Friday, March 2
8.2
15, one other, Bonus 1, Bonus 2
Friday, March 23
8.4
32, Bonus: 35 a,b,c
Friday, April 6
Debnath and Mikusinski
3.7, 3.10, 3.15
Friday, April 20
Debnath and Mikusinski
3.26, 3.29
Friday, April 27
Debnath and Mikusinski
3.41, 3.48, 3.51, 3.61 (choose 3)
Thursday, March 3
Here are the Bonus questions:
  1. If a topological space satisfies separation axiom T1 then are limits of sequences necessarily unique? If so, prove it. If not, give an example of a T1 space and a sequence which has multiple limits.
  2. Show that if the cardinality of point set X is n and X is T1, then X has the discrete topology. HINT: Use Theorem 8.6 and induction.

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