Mathematics

**MTH 503: Functional Analysis (4)**

**Pre-requisites (Desirable)**: MTH 404: Measure and Integration

*Learning Objectives*:

Functional analysis is the branch of mathematics concerned with the study of spaces of functions. This course is intended to introduce the student to the basic concepts and theorems of functional analysis with special emphasis on Hilbert and Banach Space Theory. This gives the basics for more advanced studies in modern Functional Analysis, in particular in Operator Algebra Theory and Banach Space Theory.

*Course Contents*:

- Normed Linear spaces, Bounded Linear Operators, Banach Spaces, Finite dimensional spaces, Quotient Spaces
- Hilbert spaces, Riesz Representation Theorem, Orthonormal sets, Bessel's Inequality, Parseval's Identity, Fourier Series
- Dual Spaces, Dual of L
^{p}spaces , Hahn-Banach Extension Theorem, Applications - Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle
- Weak and Weak-* topologies, Hahn-Banach Separation Theorem, Alaoglu's Theorem, Reflexivity
- Compact Operators, Adjoint of an operator, Spectral theorem for Compact Self-Adjoint operators
- (If time permits) Banach Algebras, Ideals and Quotients, Gelfand-Mazur Theorem, Fredholm Alternative, Fredholm Operators, Atkinson's theorem

*Suggested Books*:

- J.B. Conway, A Course in Functional Analysis, 2nd Ed., (Springer-Verlag, 1990)
- S. Kesavan, Functional Analysis, TRIM 52, Hindustan Book Agency
- B.V. Limaye, Functional Analysis, 2nd Ed., (New Age International, 1996)
- Martin Schechter, Principles of Functional Analysis, 2nd Ed., Graduate Studies in Mathematics, AMS
- P.D Lax, Functional Analysis, (Wiley, 2002)
- W. Rudin, Functional Analysis, 2nd Ed., (Tata McGraw-Hill, 2006)

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