Functional Analysis

Syllabus Details

• Unit I: Fundamentals of metric spaces, completion of metric spaces, normed spaces, Holder’s inequality, Minkowski’s inequality.
• Unit II: $l^p$-spaces, Banach spaces. Equivalence of norms, Riesz lemma, bounded linear maps, operator norm, separability.
• Unit III: Hahn-Banach theorem, dual of normed spaces, reflexivity, open mapping and closed graph theorems, uniform boundedness principle.
• Unit IV: Inner product spaces, Hilbert spaces, examples, projection theorem. Bessel’s inequality, existence of Schauder basis for separable Hilbert spaces, Riesz representation theorems.
• Unit V: Compact operators, finite rank operators, properties of self adjoint compact operators, spectral theorem.

Course Learning Objective

1. Explain the notions of completeness, separability, reflexivity, duality of normed spaces and apply them to solve elementary problems.
2. Apply the properties of bounded linear maps between Banach spaces/ Hilbert spaces to solve problems.
3. Explain the main results of functional analysis like Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem, closed graph theorem and apply them to solve problems.
4. Apply the special properties that Hilbert Spaces possess (like existence of orthonormal basis, Riesz representation theorem etc) to solve problems.
5. Explain compact operators, spectral theorem for self adjoint compact operators and solve problems involving these concepts.

Recommended Books

1. Kreyszig, Erwin, Introductory Functional Analysis with Applications, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989.
2. Limaye, Balmohan V., Functional Analysis, second edition, New Age International Publishers Limited, New Delhi, 1996.
3. Kesavan, S., Functional Analysis, Trim series, Hindustan Book Agency, 2009.