Module I (20 hours)
Normed spaces, Properties of normed spaces, Banach spaces, Bounded and continuous linear operators, Space of continuous linear operators, Hahn-Banach theorem and its applications, Open mapping and closed graph Theorem, Uniform Boundedness Theorem, Dual spaces, Reflexive spaces, Adjoint operator, Weak Topology and Banach-Alaoglu Theorem, Spectrum and Gelfund- Mazur Theorem.

Module II (20 hours)
Inner product spaces, Hilbert spaces, Orthonormal bases, Orthogonal complements and orthogonal projections, Riesz representation theorem, Adjoint operator, Self-adjoint, unitary and normal operators, Compact operator, Spectrums of bounded operators, Spectral Theorem.

To cover the results and concepts which are building blocks of the theory of Banach and Hilbert spaces with some applications.

CO1. Students will be able to understand the relation between algebraic and metric structure.

CO2. Students will identify the geometrical significance of Hilbert spaces.

CO3. They will learn about the spectral theory of bounded operators.

CO4. Students will be introduced to approximation theory.

CO5. Also, they will see the applications of the learned concepts to fixed point theory etc.

B. V. Limaye, Functional Analysis, New Age International

S. Ponnusamy, Foundations of Functional Analysis, Narosa Publishing House

E. Kreyszig, Introductory Functional Analysis with Applications, Wiley

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