Course Title: FUNCTIONAL ANALYSIS
Level: Graduate Course
Type: Elective
No. of Credits: 4
Pre-requisites: None
Course Rationale
Functional analysis is mainly concerned with the study of vector spaces and operators acting upon them. It provides powerful tools in handling several problems in applied mathematics and theoretical physics. It is also basic for the understanding and development of very many other mathematical theories like the Theory of Partial Differential Equations and the Theory of Operators.
Course Description
This course aims at familiarizing the student with the basic concepts, principles and methods of functional analysis and its applications. The principles learnt from basic calculus and linear algebra will be developed further to the more general setting of abstract infinite-dimensional vector spaces. Students will therefore be expected to have a solid background in undergraduate calculus, real analysis, and liner algebra.
Students will be introduced to the notion of vector spaces and the distance between vectors, as well as to continuous maps between such vector spaces. This interplay between the algebraic and analytic setting gives rise to many interesting and useful results, which have a wide range of applicability to diverse mathematical problems, such as from numerical analysis, differential and integral equations, optimization and approximation theory.
The first part of the course is devoted to a short introduction in the theory of metric spaces and to a detailed study of normed and Banach spaces and in particular to the analysis of linear operators acting upon them. The second part of the course deals with Hilbert spaces and linear operators upon them, since they play a fundamental role in applied mathematics. Finally, we look at some fundamental theorems for normed and Banach spaces such as the Hahn-Banach theorem for complex vector spaces and normed spaces and its application to bounded linear functionals; the uniform boundedness theorem, and the closed Graph theorem.
Learning Outcomes
Upon successful completion of the course, students will be able to:
- Outline the basic concepts of a metric space
- Demonstrate the properties of Normed linear spaces and Banach spaces
- Provide the fundamental theorems for normed and Banach spaces
- Describe the geometry and properties of inner product spaces
- Provide suitable representations of functionals on Hilbert spaces
- Recognize the Legendre, Hermite and Laguerre polynomials
- Utilize Zorn's lemma
- Apply the Hahn-Banach theorem for complex vector spaces and normed spaces
- State and apply the Baire Category theorem and the Uniform Boundedness theorem
- Use the Open Mapping theorem and the Closed Graph theorem
- Give examples to illustrate the use of the spectral theory of Linear Operators in normed spaces
- Start their own research projects in modern applications of functional analysis
Content (48 hours)
- Metric spaces: definition of metric space, examples; elementary topology: open sets, closed sets, neighborhood, topological space; convergence, Cauchy sequence, completeness; completion of metric spaces. (10 hours)
- Normed spaces and Banach spaces: vector space; normed space, Banach space; finite dimensional normed spaces and subspaces, compactness and finite dimension; linear operators; bounded and continuous linear operators; linear functionals; linear operators and functionals on finite dimensional spaces; normed spaces of operators, dual space. (10 hours)
- Hilbert spaces: inner product space, Hilbert space; further properties of inner product spaces; orthogonal complements and direct sums; orthonormal sets and sequences; series related to orthonormal sequences and sets; total orthonormal sets and sequences; Legendre, Hermite and Laguerre polynomials; representation of functionals on Hilbert spaces; adjoint operator; self-adjoint, unitary and normal operators. (10 hours)
- Fundamental theorems for normed and Banach spaces: Zorn's lemma; Hahn-Banach theorem for complex vector spaces and normed spaces; application to bounded linear functionals; adjoint operator; reflexive spaces; Baire Category theorem, Uniform Boundedness theorem; strong and weak convergence; convergence of sequences of operators and functionals; application to summability of sequences; Open Mapping theorem; closed linear operators, Closed Graph theorem. (10 hours)
- Special topics: In Spectral Theory of Linear Operators in normed spaces such as the spectrum and resolvent of a linear operator and their properties. (8 hours)
Teaching Methodology
Lectures: Three (3) lectures each week (50 minutes each).
Tutorials: One (1) weekly tutorial session (50 minutes of problem solving, based on theory covered during lectures).
Assessment
Coursework Mark: 40% - Two 15% Coursework examinations and 10% Assignments (*based on four assignments given during the semester)
Final Examination: 60% (one 3-hour written paper)
Course Calendar
Week |
Lecture/Tutorial Topic |
Assignments |
---|---|---|
1 |
Brief introduction to Functional Analysis and its applications. Metric spaces: definition of metric space, examples. elementary topology: open sets, closed sets. |
- |
2 |
Metric spaces: elementary topology: neighborhood, topological space; convergence; Cauchy sequence, completeness. |
Assignment #1 |
3 |
Metric spaces: completion of metric spaces. Normed spaces and Banach spaces: vector space; normed space, Banach space; finite dimensional normed spaces and subspaces. |
- |
4 |
Normed spaces and Banach spaces: compactness and finite dimension; linear operators; bounded and continuous linear operators; linear functionals. |
Assignment #2 |
5 |
Normed spaces and Banach spaces: linear operators and functionals on finite dimensional spaces; normed spaces of operators, dual space. |
- |
6 |
Hilbert spaces: inner product space, Hilbert space; further properties of inner product spaces; orthogonal complements and direct sums. |
Coursework Exam #1 |
7 |
Hilbert spaces: orthonormal sets and sequences; series related to orthonormal sequences and sets; total orthonormal sets and sequences; Legendre, Hermite and Laguerre polynomials. |
- |
8 |
Hilbert spaces: representation of functionals on Hilbert spaces; adjoint operator; self-adjoint, unitary and normal operators. Fundamental theorems for normed and Banach spaces: Zorn's lemma; Hahn-Banach theorem for complex vector spaces and normed spaces. |
Assignment #3 |
9 |
Fundamental theorems for normed and Banach spaces: application to bounded linear functionals; adjoint operator; reflexive spaces; Baire Category theorem, Uniform Boundedness theorem; strong and weak convergence. |
- |
10 |
Fundamental theorems for normed and Banach spaces: convergence of sequences of operators and functionals; application to summability of sequences; Open Mapping theorem; closed linear operators, Closed Graph theorem. |
Coursework Exam #2 |
11 |
Special topics: In Spectral Theory of Linear Operators in normed spaces such as the spectrum and resolvent of a linear operator and their properties |
Assignment #4 |
12 |
Special topics: In Spectral Theory of Linear Operators in normed spaces such as the spectrum and resolvent of a linear operator and their properties |
- |
13 |
Revision |
- |
Suggested Texts / References
Text Books:
- Kreyszig E., Introductory Functional Analysis with Applications, John Wiley & Sons, 1989.
- Simmons G.F., Introduction to Topology and Modern Analysis, International Student Edition, Mc-Graw Hill Kogakusha Ltd., 1983.
Reference Books:
- Limaye B.V., Functional Analysis, New Age International Ltd., Publishers, 2nd Edition ,1996.
- Coffman C. and Pedrick G., First Course in Functional Analysis, Prentice-Hall of India, New Delhi, 1995.
- Bollobas B., Linear Analysis, Cambridge University Press, Indian Edition, 1999.
- Nair M.T., Functional Analysis: A First course, Prentice Hall of India, 2002.
- Reed, M., and Simon, B. Methods of Modern mathematical Physics, vol.1: Functional Analysis, Academic Press, Rev Enl Su edition, 1981.
- J. B. Conway, A Course in Functional Analysis (2nd Edition), Springer New York, 2010.