Functional Analysis Course | Postgraduate Mathematics | Prof. S. Kesavan
Course Details
| Exam Registration | 48 |
|---|---|
| Course Status | Ongoing |
| Course Type | Core |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 17 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Master the Language of Modern Mathematics: A Deep Dive into Functional Analysis
Functional Analysis stands as one of the cornerstones of modern mathematical thought, providing the essential framework for numerous advanced fields. If you are a postgraduate mathematics student looking to solidify your understanding of this critical subject, a structured, expert-led course is invaluable. We are pleased to detail a comprehensive 12-week course in Functional Analysis, designed and taught by the distinguished Prof. S. Kesavan.
About the Instructor: Learn from an Eminent Mathematician
Prof. S. Kesavan brings a wealth of knowledge and experience to this course. A retired Professor from the prestigious Institute of Mathematical Sciences (IMSc), Chennai, his academic journey includes a doctoral degree from the Université Pierre et Marie Curie (Paris VI), France. His primary research interests lie in Partial Differential Equations, a field deeply intertwined with Functional Analysis.
An author of five books and a Fellow of both the Indian Academy of Sciences and the National Academy of Sciences, India, Prof. Kesavan has also held significant administrative roles, including Deputy Director of the Chennai Mathematical Institute and Secretary for the International Mathematical Union's Commission for Developing Countries. Learning from an instructor of this caliber ensures you gain insights that are both profound and practical.
Course Overview: Building a Strong Foundation
This course is meticulously designed to cover all fundamental aspects of a standard Master's level curriculum in Functional Analysis. It moves logically from basic concepts to advanced theorems, emphasizing their applications—particularly in the modern study of partial differential equations. A notable highlight is the detailed treatment of weak topologies, a crucial concept for advanced work in analysis.
Who Should Enroll?
Intended Audience: This course is ideally suited for MSc (Mathematics) students and above, including early-stage PhD researchers and professionals seeking to refresh their core analysis skills.
Prerequisites: Preparing for Success
To fully benefit from this course, students should have a solid grounding in:
- Real Analysis
- Topology
- Linear Algebra
- Measure Theory (Desirable)
Detailed 12-Week Course Layout
| Week | Topics Covered |
|---|---|
| Week 1 | Normed linear spaces, examples. Continuous linear transformations, examples. |
| Week 2 | Continuous linear transformations. Hahn-Banach theorem-extension form. Reflexivity. |
| Week 3 | Hahn-Banach theorem-geometric form. Vector valued integration. |
| Week 4 | Baire’s theorem, Principle of uniform boundedness. Application to Fourier series. Open mapping and closed graph theorems. |
| Week 5 | Annihilators. Complemented subspaces. Unbounded operators, Adjoints. |
| Week 6 | Weak topology. Weak-* topology. Banach-Alaoglu theorem. Reflexive spaces. |
| Week 7 | Separable spaces, Uniformly convex spaces, applications to calculus of variations. |
| Week 8 | L^p spaces. Duality, Riesz representation theorem. |
| Week 9 | L^p spaces on Euclidean domains, Convolutions. Riesz representation theorem. |
| Week 10 | Hilbert spaces. Duality, Riesz representation theorem. Application to calculus of variations. Lax-Milgram lemma. Orthonormal sets. |
| Week 11 | Bessel’s inequality, orthonormal bases, Parseval identity, abstract Fourier series. Spectrum of an operator. |
| Week 12 | Compact operators, Riesz-Fredholm theory. Spectrum of a compact operator. Spectrum of a compact self-adjoint operator. |
Key Textbooks and References
To complement the lectures, the following texts are recommended:
- Primary Text: Kesavan, S. Functional Analysis, TRIM 52, Hindustan Book Agency. (The instructor's own book, ensuring perfect alignment with the course material).
- Supplementary Reading: Simmons, G. F. Introduction to Topology and Modern Analysis, McGraw-Hill.
Why This Course is Essential for Your Mathematical Journey
Functional Analysis is more than just a subject; it is the language in which much of modern mathematics and its applications are written. From quantum mechanics to optimization theory, a firm grasp of Banach and Hilbert spaces, duality, and spectral theory is indispensable. This course, with its structured approach from basics to significant results like the Hahn-Banach Theorem, Open Mapping Theorem, and Riesz-Fredholm Theory, will equip you with the tools needed to tackle advanced problems in analysis, differential equations, and beyond.
Under the guidance of Prof. Kesavan, you will not only learn the theorems but also appreciate their power through targeted applications. Enroll today to take a significant step towards mastering the elegant and powerful world of Functional Analysis.
Enroll Now →