Details of MA4102 (Autumn 2019)

Level: 4 Type: Theory Credits: 4.0

Course CodeCourse NameInstructor(s)
MA4102 Functional Analysis Shibananda Biswas

MA4102 : Functional Analysis

Normed Linear Spaces: Definitions, Banach Spaces, Hilbert spaces, non-compactness of the unit ball in infinite dimensional normed linear spaces, quotient spaces.

Linear Maps: Boundedness and continuity, linear functionals. isometries, Mazur-Ulam theorem on isometries.

Completeness: Banach-Steinhaus theorem, open mapping theorem and closed graph theorem.

Convexity: Hahn-Banach extension theorem, complex Hahn-Banach theorem, separation of convex sets, applications.

Duality: Dual spaces, Riesz representation theorem, reflexivity, Eberlain-Schmulian theorem, weak topologies, weak convergence, weak compactness, Banach-Alaoglu theorem, Krein-Milman theorem, adjoints and compact operators.

Hilbert Spaces: Bessels inequality, complete systems, Gram-Schmidt orthogonalization, Parsevals identity, projections, orthogonal decomposition, bounded linear functionals in Hilbert spaces.

Spectral Theory: Spectrum, Fredholm theory of compact operators, spectral theory of compact self-adjoint operators, minimax principle, application to integral operators.

Analysis IV (MA3204) and Topology (MA3201).

Suggested Texts/Reference Books:

1. Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer.

2. Bollabs, B., Linear Analysis: An Introductory Course, Cambridge University Press.

3. Conway, J.B., A Course in Functional Analysis (2nd Edition), Springer-Verlag.

4. Eidelman, Y., Milman, V. and Tsolomitis, A., Functional Analysis: An Introduction,American Mathematical Society.

5. Kesavan, S., Functional Analysis, Hindustan Book Agency.

6. Lax, P.D., Functional Analysis, Wiley-Interscience.

7. Limaye, B.V., Functional Analysis, New Age Publishers.

8. Rudin, W., Functional Analysis, McGraw-Hill.

9. Simmons, G.F., Topology and Modern Analysis, Tata McGraw-Hill.

Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 1 Not Allowed
2 IP 3 Core
3 IP 5 Not Allowed
4 MR 1 Not Allowed
5 MR 3 Not Allowed
6 MS 7 Core
7 RS 1 Elective
8 RS 2 Elective