### MA5450 Functional Analysis

#### Course Details

Unit I: (14 lectures)

Normed linear spaces, Banach spaces; Classical examples: C([0,1]), l_{p}, c, c_{0}, c_{00}, L_{p}[0,1]; Continuity and boundedness of linear operator; Quotient spaces; Finite dimensional normed spaces; Riesz lemma, (non)compactness of unit ball; Seperability with examples;

Unit II: (10 lectures)

Hahn Banach extension theorem, Open mapping theorem, Closed graph theorem, Uniform boundedness principle.

Unit III: (16 lectures)

Hilbert spaces, Projection theorem; Orthonormal basis, Bessel inequality, Parseval equality; Dual, Duals of classical spaces-c_{0}, l_{p}, L_{p}[0,1]; Riesz representation theorem, Adjoint of an operator; Double dual, Weak and weak* convergence;

#### Course References:

Text Books:

1. M. Fabian, P. Habala, P. Hajek, V. M. Santalucia, J. Pelant and V. Zizler, Functional analysis and
infinite-dimensional geometry. (Canadian Math. Soc, Springer 2001).

2. M. T. Nair, Functional analysis. (PHI-Learning, New Delhi, Fourth Print 2014).

Reference Books:

1. B. Bollobas, Linear analysis (Cambridge Univ. Press 1999).

2. J. Conway, A course in functional analysis. (Springer 2007).

3. C. Goffman and G. Pedrick, A first course in functional analysis, (Prentice-Hall 1974).

4. P. D. Lax, Functional analysis (Willey interscience 2002).

5. B.V Limaye, Functional analysis (New Age International 1996).

6. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol I. (Academic press, 1980).