This course extends methods of linear algebra and mathematical analysis to spaces of functions, in which the interaction between algebra and mathematical analysis allows powerful methods to be developed. The course will be mathematically sophisticated and will use ideas both from linear algebra and analysis.

Prerequisite

Complex Analysis (MATH-306)

Learning Outcomes

After taking this course the students should be able to independently prove and thoroughly explain central theorems of functional analysis, should be able to apply Hilbert space theory, Riesz representation theorems and theory of operators in problem solving.

COURSE CONTENTS

1. Metric Spaces
2. Cauchy’s sequences and examples
3. Completeness of metric space
4. Completeness proofs
5. Normed linear Spaces
6. Banach Spaces
7. Quotient Spaces
8. Continuous and bounded linear operators
9. Linear functional
10. Linear operator and functional on finite dimensional Spaces.
11. Hilbert Spaces: Inner product Spaces, Hilbert Spaces (definitions and examples)
12. Conjugate spaces
13. Representation of linear functional on Hilbert space
14. Reflexive spaces.

Recommended Book

1. E. Kreyszig, Introduction to Functional Analysis with Applications, (John Wiley and sons, 1989)

Suggested Books

1. Curtain R.F., Pritchard A.J.. Functional Analysis in Modern Applied Mathematics, 1st ed. NY: Academic Press, 1977.

2. Friedman A. Foundations of Modern Analysis. 1st ed.  Dover, 1982.

3. Rudin W. Functional Analysis. 1st ed. NY: McGraw Hill, 1973.

ASSESSMENT CRITERIA

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Mid-Term Exam:   30

Final-Term Exam: 50