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FUNCTIONAL ANALYSIS

By DR. MANTU SAHA   |   THE UNIVERSITY OF BURDWAN
Learners enrolled: 295
Functional Analysis, which yields a unifying framework for any areas of mathematics, namely theory of functions, theory of measure and integration, theory of approximations, theory of distribution, Fourier analysis, theory of Sobolev spaces,etc. plays a vital role in pure and applied sciences. It leads to solve different types of problems in mathematics, physics, mechanics, operations research, mathematical economics and many other disciplines. It is therefore, the study of functional analysis is very much essential to pursue mathematics, especially applied mathematics and applied sciences. 
The course , Functional Analysis , based on lecture materials in my post graduate teaching learning class has been designed to cater to the need of students who are yet to be exposed to the subject as well as post graduate students in various national universities and abroad. Based on the evelopment of this subject, the content of the materials on this course has been designed for the need of providing clarity to our students as well as to present it within the comprehension of the subject. As a result of all this, the text materials are presented in a simple and lucid language with illustrative familiar examples. 
The texts have been arranged in different ten chapters with thirty five modulus and sixty seven videos with a view to begin a chapter on preliminaries and providing a sequence of definitions and theorems. Some modules in each chapter are intended to establish a uniform notation and cover the background material in real analysis, linear algebra and metric spaces. It is followed by important information on functional analysis; namely, normed linear spaces and Banach spaces, bounded linear operators, and bounded linear functionals. 
Each module of each chapter of this subject starts with a brief introduction to invoke interest of the students in what follows in the texts. Each module is profusely illustrated with familiar examples to make the students understanding easily on the underlying concept of this course and in solving concrete problems. Most of the theoretical portions have also been introduced to the students by citing examples. 
This, I feel, should help in easy comprehension of this subject. Wherever essential, students are provided with hints to solve the problems. The problems for all the chapters graded in a proper way are presented in ‘Self Check Exercises’. Also a list of References ‘Learn More’ has also been provided. 
Without claiming originality of results we do claim simplicity and lucidity of presentation coupled with comprehensive of the material.

Summary
Course Status : Ongoing
Course Type : Core
Duration : 15 weeks
Category :
  • Mathematics
Credit Points : 4
Level : Postgraduate
Start Date : 15 Jan 2024
End Date : 28 Apr 2024
Enrollment Ends : 29 Feb 2024
Exam Date : 25 May 2024 IST
Shift - II :

3PM - 6PM

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

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Course layout

WEEK No.

Chapter No.

Chapter Name 

Module

Module Name

Week 1

1

Metric Spaces

1

Fundamental Inequalities

 

 

 

2

Some Properties On Metric Spaces

 

 

 

3

Metric Subspaces

 

 

 

 

 

Week 2

2

Fundamental Theorems For Metric Spaces

1

Completion Of Metric Spaces

 

 

 

2

Compactness Of c[a,b]

 

 

 

3

Fixed Point Of Contraction Mappings

 

 

 

 

 

Week 3

3

Normed Linear Spaces and Banach Spaces

1

Linear Spaces

 

 

 

2

Basic Properties Of A Normed Linear Space

 

 

 

3

Examples Of Banach Spaces

 

 

 

 

 

Week 4

3

Normed Linear Spaces And Banach Spaces

4

Some Important Results On Normed Linear Spaces and Banach spaces

 

4

Characterization Of Banach Spaces

1

Convex Sets

 

 

 

2

Equivalent Norms And Series In Banach spaces

 

 

 

 

 

Week 5

4

Characterization Of Banach Spaces

3

Quotient Spaces

 

5

Bounded Linear Operators

1

Basic Results On Bounded Linear Operators

 

 

 

2

Norm Of Bounded Linear Operators

 

 

 

 

 

Week 6

5

Bounded Linear Operators

3

Convergence Of Bounded Linear Operators

 

6

Fundamental Theorems For Bounded Linear Operators

 

 

 

 

 

 

1

Open Mapping Theorem

 

 

Week 7

6

Fundamental Theorems for Bounded Linear Operators

2

Closed Graph Theorem

 

 

 

3

Extension Of Bounded Linear Operators

 

 

 

 

 

Week 8

 

Revision and Assignment Week 

 

 

 

 

 

 

 

Week 9

7

Fundamental Theorems For Bounded Linear Functionals

1

Linear Functionals

 

 

 

2

Hahn Banach Theorem

 

 

 

3

Applications Of Hahn Banach Theorem

 

 

 

 

 

Week 10

8

Conjugate Spaces

1

First Conjugate Spaces

 

 

 

2

Second Conjugate spaces

 

 

 

3

Strong Convergence And Weak Convergence Of A Sequence Of Operators

 

 

 

 

 

Week 11

8

Conjugate Spaces

4

Conjugate Operators On Normed Linear Spaces

Week 11

9

Inner Product Spaces And Hilbert Spaces

1

Inner Product Spaces

 

 

 

2

Orthogonal And Orthonormal Vectors

 

 

 

 

 

Week 12

9

Inner Product Spaces And Hilbert Spaces

3

Some Fundamental Results On Inner Product Spaces

 

 

 

4

Some Results On Hilbert Spaces

 

 

 

 

 

Week 13

9

Inner Product Spaces And Hilbert spaces

5

Series In Hilbert Spaces And Isometric Isomorphism Between Hilbert spaces

 

10

Classification Of Operators Over Hilbert Spaces

1

Adjoint Operators And Algebra Of Adjoint Operators

 

 

 

 

 

Week 14

10

Classification Of Operators Over Hilbert Spaces

2

Self Adjoint Operators Over Hilbert Spaces And Its Eigen Values And Eigen Vectors

 

 

 

3

Normal Operators And Unitary Operators

 

 

 

4

Projection Operators

 

 

 

 

 

Week 15

 

Assessment and overview of the course

 

 

Books and references

1. E. Kreyszig - Introductory Functional Analysis with Applications, John Wiley and Sons, 1989.

2. A.E. Taylor - Functional Analysis, John Wiley and Sons, New York, 1958.

3. P.K Jain and O.P. Ahuja - Functional Analysis, New Age International Pvt. Limited, 2011

4. G. Bachman and L. Narici -Functional Analysis, Academic Press, 1966.

5. S.K. Berberian - Introduction to Hilbert spaces, Oxford University Press, 1961.

6. D. Somasundaram - A first course in functional Analysis, Narosa Publishing House, 2006.

Instructor bio

DR. MANTU SAHA

THE UNIVERSITY OF BURDWAN

Personal Profile: 
Name: DR. MANTU SAHA
Designation: PROFESSOR OF MATHEMATICS
DEPARTMENT OF MATHEMATICS
THE UNIVERSITY OF BURDWAN
BURDWAN
 Qualifications: M.Sc. (First Class First (Pure Stream), Gold Medalist), PGDCA, Ph.D.
 Teaching Experience: More than 25 Years.
 Research Experience: More than 32 Years.

 Publications:
 Research Papers: 101 nos. 
 Research Monograph: 01 no.
 
 Research Project:
(i) Completed: 01 no.
(ii) Completed Departmental Project UGC (DSA-SAP (I)):01 no., as Deputy Coordinator (2012-2017)].
 Research Guidance:
 M.Phil. Awarded: 01 no.
 Ph.D.:
(i) Awarded: 07 nos. 
(ii) Working: 01 no.
 Field of Specialization: Functional Analysis and Operator Theory.
 Area of Research: Fixed Point Theory and Its Applications
 Topic of Doctoral Thesis: A Study on Fixed Points of Mappings over a Quasi Metric Space and Over a Banach Space with a Probability Measure.
 Reviewing of Research paper(s): Several research papers submitted to International Journals (Referred/Peer Reviewed/Springer) and Reviewer Mathematical Reviews of American Mathematical Society.

Administrative Experience: 
 Management Trainee in WBSEB (12.07.1994 -11.07.1995).
 Assistant Engineer (U) in WBSEB (12.07.1995 -15.11.1997).
 Head of the Department of Mathematics, The University of Burdwan (19.06.2008-18.06.2010)

Academic Involvement: 
 Invited lectures delivered in seminars/ conferences/ symposiums/workshops:
 International: 05 nos.
 National /State /Regional: 28 nos.
 Participation and Presentation of Research papers in seminars/conferences:
 International: 04 nos.
 National: 07 nos.
 Participation in Training Courses: 03 nos.
 Participation in Workshops: 03 nos.
 Participation in Seminars/ Symposiums /Conferences
 Participation: 10 nos.
 Organized Seminars/Conferences as Convener /Joint Convener: 03 nos.
 Organized Refresher Courses as Coordinator: 03 nos.

 Course Taught:
 (Post-Graduation) Functional Analysis, Advanced Functional Analysis, Mathematical Logic, Graph Theory, Algebra, Topology, Complex Analysis, Operator Theory and its Applications, Transformed calculus.
 (Graduation) Algebra, Calculus, Analysis, Geometry, Numerical Analysis, Probability and Statistics.

Awards/Honours: 
 Awarded with Professor Sudhodhan Ghosh Memorial Lecture Award by Calcutta Institute of Theoretical Physics(2013)
 Got the position in World Scientist University Ranking (2021,2022) [AD Scientific Index]

Professional Activities:
 Former Director, UGC-HRDC, The University of Burdwan
 Involved as a member in NAAC Peer Team
 Reviewer in Mathematical Reviews, American Mathematical Society
 Content writer of Post Graduate Course in Functional Analysis and Advanced Functional Analysis under Directorate of Distance Education, The University of Burdwan
 Content writer of Post Graduate Course on Functional Analysis[ Mathematics] in e-PG Pathshala Programme, MHRD, Govt. of India
 Holding a Life Membership of Calcutta Mathematical Society
 Holding a Life Membership of Indian Science Congress Association
 Holding a life Membership of Indian Mathematical Society

Research Profile Information 
:: VIDWAN Info.: https://vidwan.inflibnet.ac.in/profile/195913
:: Google Scholar Info.: https://scholar.google.com/citations?user=2y31ZA4AAAAJ&hl=en
:: ORCID Info.: https://orcid.org/0000-0003-0128-2268
:: Research gate info: https://www.researchgate.net/profile/Mantu-Saha
:: Additional information: University website: https://buruniv.ac.in

Declaration:
I do hereby declare that the information given above is true to the best of my knowledge and belief.

Course certificate

30 Marks will be allocated for Internal Assessment and 70 Marks will be allocated for end term proctored examination
Securing 40% in both separately is mandatory to pass the course and get Credit Certificate.


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