### Functional Analysis [MATH-506]

MATH-506                 Functional Analysis

Description

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

Learning Outcomes

This course will enable students to connect the algebraic concepts with analysis. Students will be able to understand the concept of metric spaces and their completeness. The concept of normed and Banach spaces gives relation between vector spaces and metric spaces. Moreover, students will learn the scope of inner product and Hilbert spaces.

Contents

1. Metric Spaces
2. Convergence
3. Cauchy’s sequences and examples
4. Completeness of metric space
5. Completeness proofs
6. Banach Spaces: Normed linear Spaces, 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 Books

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

Suggested Books

1. N. Dunford and J.T. Schwartz, Linear Operators (part-1 General theory), (Interscience publishers, New York, 1958)

2. Seymour Lipschutz, Outline of General Topology,( 2011)

Assessment Criteria

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

Mid-Term Exam:   30

Final-Term Exam: 50

### Key Dates and Time of Class Meeting

Wednesday                                              08:00 AM-09:00 AM (Reg)

Thursday                                                  12:30 PM-02:00 PM (SS)

Friday                                                       09:30 AM-11:00 AM (Reg)                      02:00 PM-03:30 PM (SS)

Commencement of Classes                                                   March 02, 2020

Mid Term Examination                                                            April 27 to May 04, 2020

Final Term Examination                                                          June 22-26, 2020

Declaration of Result                                                              July 03, 2020