Aktosun, Fall 2023, Math 5327

Instructor: Tuncay Aktosun, PKH 461, (817) 272-1545, aktosun@uta.edu

Course: Functional Analysis I, Math 5327, Section 001, Course number 86300

Prerequisite: Math 5317 (Real Analysis I)

Office Hours: 5:00 pm-6:00 pm, Tuesday and Thursday and by appointment

Classroom: SWSH 429 (New Nursing Building 429)

Classtime: 3:30 pm-4:50 pm, Tuesday and Thursday

Textbook: There is not an official textbook for the course. For beginners without advanced training, the book by Naylor and Sell can be an excellent source to learn the basic material. The book by Kreyszig, the book by Robinson, and the book by Rynn and Youngson can also be good learning sources for beginners.

Some useful books:
N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, 2 volumes, Dover Publ., New York, 1993.
J. B. Conway, A course in functional analysis, 2nd ed., Springer, New York, 1997.
R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 1, Wiley, New York, 1989.
L. Debnath and P. Mikusinski, Introduction to Hilbert spaces with applications, 3rd ed., Academic Press, New York, 2015.
C. Goffman and G. Pedrick, A first course in functional analysis, 2nd ed., Am. Math. Soc., Providence, 2017.
L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, New York, 1982.
A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Vol. 1, Greylock Press, Rochester, NY, 1957.
A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Vol. 2, Greylock Press, Albany, NY, 1961.
E. Kreyszig, Introductory functional analysis, Wiley, New York, 1989.
A. W. Naylor and G. R. Sell, Linear operator theory in engineering and science, Springer, New York, 1982.
J. C. Robinson, Introduction to functional analysis, Cambridge University Press, New York, 2020.
M. Rosenlicht, Introduction to Analysis, Dover Publ., New York, 1986.
B. Rynne and M. A. Youngson, Linear functional analysis, 2nd ed., Springer, New York, 2007.
K. Saxe, Beginning functional analysis, Springer, New York, 2010.
M. Schechter, Principles of functional analysis, 2nd ed., Amer. Math. Soc., Providence, RI, 2001.
A. E. Taylor and D. C. Lay, Introduction to functional analysis, 2nd ed., Krieger, Malabar, FL, 1980.

Coverage: The covered material includes linear spaces, functionals, operators, Banach spaces, Hilbert spaces, and some applications. The topics to be covered are indicated in the course outline.

Grading: A letter grade will be assigned based on a class presentation. Unless otherwise indicated, all grades out of 100 will correspond to the following scale: 0 < F< 60, 60 < D < 70, 70 < C < 80, 80 < B < 90, 90 < A < 100.

Course outline
Sample problems 1
Sample problems 2

Biographies of mathematicians
Functional analysis (wikipedia)
Functional analysis (Encyclopedia of mathematics)
Functional analysis (Mathonline)
Field (wikipedia)
Lebesgue integral (wikipedia)
Metric space (wikipedia)
Banach space (wikipedia)
Hilbert space (wikipedia)
Aleph number, cardinality (wikipedia)
Function space (wikipedia)
Sequence space (wikipedia)
Cauchy sequence (wikipedia)
Operator norm (wikipedia)
Unbounded linear operators and closed linear operators (wikipedia)
Fredholm alternative (wikipedia)
Proof of the Fredholm alternative (Terry Tao)
Uniform boundedness principle (wikipedia)
Hahn-Banach theorem (wikipedia)
Open mapping theorem (wikipedia)
Closed graph theorem (wikipedia)
Spectral theorem (wikipedia)
The BLT theorem (wikipedia)
Adjoint of an operator (wikipedia)
Normal operators (wikipedia)

Tuncay Aktosun aktosun@uta.edu
Last modified: July 25, 2023