### MA 5450 Functional Analysis

#### Course Details

Unit I: Normed linear space; Banach spaces and basic properties: Heine-Borel theorem, Riesz lemma and best approximation property: Inner product space and projection theorem; Orthonormal bases; Bessel inequality and Parseval's formula; Riesz-Fischer theorem.

Unit II: Bounded operators and basic properties; Space of bounded operators and dual space; Riesz representation theorem; Adjoint of operators on a Hilbert space; Examples of unbounded operators; Convergence of sequence of operators.

Unit III: Hahn-Banach Extension theorem;Uniform boundedness principle; Closed graph theorem and open mapping theorem. Some applications.

Unit IV: Invertibility of operators; Spectrum of an operator

#### Course References:

Text Books:

1. M.T.Nair: Functional Analysis: A First Course, Prentice Hall of India, 2002(Third Printing, PH1 Learning Pvt. Ltd., 2010)

2. B.V. Limaye: Functional Analysis, Second Edition New Age International, 1996

References:

1. B.Bollabas: Linear Analysis, Cambridge University Press (Indian edition),1999.

2. E.Kreyszig: Introdiction to Functional Analysis with Applications,wiley,1989.

3. S.Ponnusamy: Foundations of Functional Analysis,Narosa, 2003.

4. G.F.Simmons:Introduction toTolpology and Modern Analysis,McGraw-hill, 1963.

5. A.E. Taylor and D.C. Lay: Introduction to Function Analysis, 2nd ed., Wiley, New York, 1980.