Functional Analysis

Course Objectives: Functional analysis is a fundamental area of pure mathematics, with countless applications to the theory of differential equations, engineering, and physics. The students will be exposed to the theory of Banach space, Hilbert spaces, linear transformations and functionals. In particular, the major theorems in functional analysis, namely, Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem and closed graph theorem will be covered.

Course Description:

Review of some basic concepts in metric spaces and topological spaces, completeness proofs and completion of metric spaces.

Normed linear spaces and Banach spaces, examples of Banach spaces, quotient spaces, equivalent norms, finite dimensional Banach spaces and compactness, bounded and continuous linear operators and linear functionals, dual space, Banach fixed-point theorem and applications, Hahn-Banach theorem and applications, uniform boundedness theorem, open mapping and closed graph theorem, weak and weak* convergence.

Inner product space and its properties, Hilbert spaces and examples, best approximation in Hilbert spaces, orthogonal complements, orthonormal basis, dual of a Hilbert space.

Operator theory, adjoint of an operator, Riesz representation theorem, self-adjoint operators, normal and unitary operators, projections, compact operators.

Course learning outcomes (CLO): On successful completion of this module, students will be able to:

1) understand the fundamentals of complete metric spaces and normed linear spaces.

2) recognize finite dimensional spaces and associated properties.

3) independently prove and thoroughly explain central theorems.

4) understand thoroughly inner product and Hilbert spaces and can provide approximations in Hilbert spaces.

5) define and thoroughly explain self-adjoint, normal and unitary operators and analyze operators from applications.

Recommended books:

1) Erwin Kreyszig, Introductory Functional Analysis with Applications, Wiley, 2007.

2) John B. Conway, A course in Functional Analysis, Springer, 2nd edition, 1990.

3) G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw Hill Education, 2017.

4) S. Kesavan, Functional Analysis, Hindustan Book Agency, 2014.

5) Peter D. Lax, Functional Analysis, John Wiley & Sons, 2002.