Math 535 Applied Functional Analysis Spring, 2008
Meetings: TR 4:30-5:45, Sci 191
Instructor: Greg Morrow
Office: Engr 271, (719) 262-3184
Hours: TR: 11:00 -11:45, 3:30-4:15, or by appointment.
E-mail: gmorrow@uccs.edu
Web: http://www.uccs.edu/~gmorrow/
Video Archives: http://www.uccs.edu/~math/video/index.php
Prerequisite: Advanced Calculus in one real variable (Math 431)
Text: Introductory Functional Analysis with Applications by Erwin Kreyszig, ISBN 0-471-50731-8, Wiley (1978).
Description: An introduction to the basic concepts, methods and applications of functional analysis. Topics covered include metric spaces, normed and Banach spaces, linear operators, Hilbert spaces, representation of functionals on Hilbert space, Hahn-Banach theorem , Uniform-boundedness theorem, weak* convergence, Open mapping theorem, Closed graph theorem, contraction mapping theorem and applications, approximation theory and applications.
Homework and Tests: Homework will be assigned and collected weekly; see the assignments and due dates on the syllabus. Late homework will not be accepted. The lowest homework score will be dropped. There will be one mid-semester test and a final test, as well as a project; see the guidelines below. The final exam will cover material primarily from the material after the second midterm test. The following percentages may be used to compute the final grade: homework, 35%; tests, 25% each, project 15%.
Grading: 90 => grade of A , 80 => grade of B or better, 70 => grade of C or better.
1/22 , 1/24 2.1-2.3 Notes 1
1/29, 1/31 2.4-2.6 Notes 2 HW I, Jan. 31
2/5, 2/7 2.7-2.9 Notes 3 HW II, Feb. 7
2/12, 2/14 2.10 , 3.1-3.2 Notes 4 HW III, Feb. 14
2/19, 2/21 3.3-3.4 Notes 5 HW IV, Feb. 21
2/26, 2/28 3.5 -3.7 Notes 6 HW V, Feb. 28
3/4, 3/6 3.8-3.10 Notes 7 HW VI, Mar .6
3/11, 3/13 4.1-4.3 Exam 1 Notes 8 HW VII, Mar. 13
3/18, 3/20 Exam I, covering 2.1-3.10, is due on 3/20.
3/25, 3/27 Spring Break
4/1, 4/3 4.4-4.6 Notes 9 HW VIII, April 3
4/8, 4/10 4.7-4.9 Notes 10 HW IX, Apr. 10
4/15, 4/17 4.11-4.13 Notes 11 HW X, Apr. 22
4/22, 4/24 5.1 -5.5 Final Exam HW XI, May 1
4/29, 5/1 6.1-6.6 Notes 12 Notes 13 projects
5/6, 5/8 11.1-11.3 projects
5/13 Final Test Due: 5/13, 5:00 pm
II. 2.4: 2, 6; 2.5: 4;
III. 2.6: 4, 12, 14 (EC); 2.7: 6, 7, 8, 11; 2.8: 2, 3, 6;
IV. 2.9: 4, 6. 2.10: 6, 8; 3.1: 3, 6 11.
V. 3.2: 4; 3.3: 1, 6, 8; 3.4: 6, 8, 9.
VI. 3.5: 6, 8; 3.6: 4, 10; 3.7: 1, 3 (EC), 6 (EC).
VII. 3.8: 3, 7; 3.9: 2, 4, 10; 3.10: 6.
VIII. 4.1: 4, 6; 4.2: 6; 4.3: 9 (EC), 12, 14 (for a real normed space).
IX. 4.5: 8, 9; 4.6: 2, 6, 8.
X. 4.7: 6, 8, 10; 4.8: 4; 4.9: 4.
XI. 4.12: 6, 8; 4.13: 8, 9 (extra credit. Revise 4.13.9 as follows since the only compact normed space is Y={0}. Assume that T:X->Y is closed and that for each convergent sequence (x_n) in X the sequence (y_n)=(Tx_n) has a convergent subsequence in Y. Then prove that T is bounded.)
Project Guidelines:
Submit a 10 pages
typed paper. Present paper in 20-25 minutes to the class. Include certain
examples (beyond those worked out in the text), such as problems from the text,
to illustrate the subject. Consultation with Morrow or other sources is
appropriate. Projects focus on one of the following combinations of Sections
from the text:
(a) 5.1,5.2; (b) 5.1,
5.3; (c) 5.1, 5.4; (d) 6.1-6.2, 6.3; (e) 6.1-6.2, 6.4 (assume 6.3); (f)
6.1-6.2, 6.5; (g) 6.1-6.2, 6.6; (h) 11.1-11.3.