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Welcome and have a look at an area of Math that can easily be described as the unification of ideas from several other core areas…
Course Code: ME010304
Course Credit: 4
Course Type: Core
Functional Analysis plays an important role in the applied sciences as well as in mathematics itself.The impetus came from applications: problems related to ordinary and partial differential equations, numerical analysis, calculus of variations, approximation theory, integral equations, and so on.
On successful completion of this course, students will be able to appreciate how functional analysis uses and unifies ideas from vector spaces, the theory of metrics, and complex analysis.
The objectives of the course are the study of :
- The basic results associated to different types of convergences in normed spaces .
- The main properties of bounded operators on Banach spaces.
- Functionals, Dual spaces and their significance.
- Hilbert Spaces and representation of functionals on Hilbert Spaces.
- The Hahn Banach Theorems for real, complex and normed spaces.
- To define and thoroughly explain Banach and Hilbert spaces and self-adjoint operators.To decide which properties an operator has.
- To analyse operators and applications from a critical point of view .
- To independently decide if a linear space is a Banach space.
- To understand the notions of dot product and Hilbert space.
- To apply Hilbert space-theory, including Riesz’ representation theorem and critically reflect over chosen strategies and methods in problem solving.
- To be acquainted with the statement and proof of the Hahn-Banach theorems and its corollaries.
Text Book: Erwin Kreyszig, Introductory Functional Analysis with applications, John Wiley and sons, New York
1. Limaye, B.V, Functional Analysis, New Age International (P) LTD, New Delhi, 2004
2. Simmons, G.F, Introduction to Topology and Modern Analysis, McGraw –Hill, New York, 1963
3. Siddiqi, A.H, Functional Analysis with Applications, Tata McGraw –Hill, New Delhi, 1989
4. Somasundaram. D, Functional Analysis, S.Viswanathan Pvt. Ltd, Madras, 1994
5. Vasistha, A.R and Sharma I.N, Functional analysis, Krishnan Prakasan Media (P) Ltd, Meerut: 1995-96
6. M. Thamban Nair, Functional Analysis, A First Course, Prentice – Hall of India Pvt. Ltd, 2008
7. Walter Rudin, Functional Analysis, TMH Edition, 1974
The course commences with a recap of the basics of linear algebra, and topology.The basics of vector spaces are covered and are slowly built upon to reach the generalization into inner product spaces and several theorems that are considered the pillars of modern day functional analysis. Besides regular lectures, there will be sessions for problem solving, seminars and discussions too.
The internal evaluation is based on the participant’s participation, assignments, seminars, internal assessments and an external end semester exam.
The external end semester exam carries a weightage of 4 whereas the internal component carries a weightage of 1.
Examples, Completeness proofs, Completion of Metric Spaces, Vector Space, Normed Space, Banach space, Further Properties of Normed Spaces, Finite Dimensional Normed spaces and Subspaces, Compactness and Finite Dimension (Chapter 1 – Sections 1.5, 1.6; Chapter 2 - Sections 2.1 to 2.5)
Linear Operators, Bounded and Continuous Linear Operators, Linear Functionals, Linear Operators and Functionals on Finite dimensional spaces, Normed spaces of operators, Dual space (Chapter 2 - Section 2.6 to 2.10)
Inner Product Space, Hilbert space, Further properties of Inner Product Space, Orthogonal Complements and Direct Sums, Orthonormal sets and sequences, Series related to Orthonormal sequences and sets, Total Orthonormal sets and sequences, Representation of Functionals on Hilbert Spaces (Chapter 3 - Sections 3.1 to 3.6, 3.8)
Hilbert-Adjoint Operator, Self-Adjoint, Unitary and Normal Operators, Zorn’s lemma, Hahn- Banach theorem, Hahn- Banach theorem for Complex Vector Spaces and Normed Spaces, Adjoint Operators (Chapter 3 - Sections 3.9, 3.10; Chapter 4 - Sections 4.1 to 4.3, 4.5)
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