### MA7840 Analysis

#### Course Details

Convexity and Extreme Points: Topologies on linear spaces, linear functionals on topological spaces, weak topology, weak*topology, extreme points, Krein-Milman theorem. functions.

Banach Algebras: Banach algebra, complex homomorphisms on a Banach algebra, properties of spectra, spectra radius formula, Gelfand - Mazur theorem, Gelfand transform, Maximal ideal space, involutions, Gelfand - Naimark theorem.

Spectral Theory: Bounded operators on a Hilbert space, normal, self adjoint, unitary and projection operators, resolutions of the identity, spectral theorem and symbolic calculus of normal operators.

#### Course References:

Text Books:

1. W.Rudin, Functional Analysis, International series in pure and applied Mathematics, Tata-McGraw Hill edition, 2007.

2. S.David Promislow, A first course in Functional Analysis, Pure and applied Mathematics, Wiley-Interscience, 2008.

Reference:

1. M.T.Nair, Functional Analysis: A First Course, Prentice-Hall of India, New Delhi,2002.

2. Peter D.Lax, Functional Analysis, Wiley-Interscience,2002.

3. J.B.Conway, A course in Functional Analysis, GTM 96, Springer, 1985.