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.
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.
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.