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Functional Analysis

By Prof. S. Kesavan   |   IMSc
Learners enrolled: 249
Functional Analysis is a core course in any mathematics curriculum at the masters level. It has wide ranging applications in several areas of mathematics, especially in the modern approach to the study of partial differential equations. The proposed course will cover all the material usually dealt with in any basic course of Functional Analysis. Starting from normed linear spaces, we will study all the important theorems, with applications, in the theory of Banach and Hilbert spaces. One important feature of the proposed course is the detailed treatment of weak topologies. Prerequisites are familiarity with real analysis, topology and linear algebra. Knowledge of measure theory is desirable .

INTENDED AUDIENCE
MSc (Mathematics) and above
PREREQUISITES : BSc (Mathematics) Real Analysis, Topology, Linear Algebra, Measure Theory
INDUSTRIES  SUPPORT     : None
Summary
Course Status : Upcoming
Course Type : Core
Duration : 12 weeks
Start Date : 24 Jan 2022
End Date : 15 Apr 2022
Exam Date : 24 Apr 2022 IST
Category :
  • Mathematics
Credit Points : 3
Level : Postgraduate



Course layout

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 the 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-adjont operator.

Books and references

1. S. Kesavan, Functional Analysis, TRIM 52, Hindustan Book Agency, New Delhi.
2. Simmons, G. F. Introduction to Topology and Modern Analysis, McGraw-Hill, Kogakusha, International Student Edition.

Instructor bio

Prof. S. Kesavan

IMSc
S. Kesavan retired as Professor from the Institute of Mathematical Sciences, Chennai. He obtained his doctoral degree from the Universite de Pierre et Marie Curie (Paris VI), France. His research interests are in Partial Differential Equations. He is the author of five books. He is a Fellow of the Indian Academy of Sciences and the National Academy of Sciences, India. He has served as Deputy Director of the Chennai Mathematical Institute (2007-2010) and two terms (2011-14, 2015-18) as Secretary (Grants Selection) of the Commission for Developing Countries of the International Mathematical Union. He was a member of the National Board for Higher Mathematics during 2000-2019.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 24 April 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team


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