Course Details
Subject {L-T-P / C} : MA6613 : Functional Analysis with Applications { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Prof. Divya Singh
Syllabus
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.
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.
Course Objectives
- To cover the results and concepts which are building blocks of the theory of Banach and Hilbert spaces with some applications.
Course Outcomes
1. Students will be able to understand the relation between algebraic and metric structure. <br />2. Students will identify the geometrical significance of Hilbert spaces. <br />3. Students will get an idea of approximation theory and application of the learned concepts in fixed point theory etc.
Essential Reading
- B. V. Limaye, Functional Analysis, New Age International
- S. Ponnusamy, Foundations of Functional Analysis, Narosa Publishing House
Supplementary Reading
- E. Kreyszig, Introductory Functional Analysis with Applications, Wiley
- Y. Eidelman, V. Milman, A. Tsolomitis, Functional Analysis: An Introduction, American Mathematical Society