Normed linear spaces, Banach spaces, Equivalence of norms on finite dimensional spaces, Riesz lemma and characterization of finite dimensional normed spaces, Hamel basis and Schauder basis of normed spaces, Separable normed spaces.
Bounded linear operators, Continuous linear functionals, Hahn-Banach theorems (separation and extension theorems), Dual and bidual of a normed linear space, Dual of some classical spaces like c0, lp, Lp (for p≥ 1) and C(K); Reflexive spaces, weak convergence.
Uniform boundedness principle, Open mapping theorem, Closed graph theorem.
Transpose of an operator, Compact operators, Spectra of bounded linear operators and compact operators.
Hilbert spaces, Bessel’s inequality, Orthonormal basis, Separable Hilbert space, Orthogonal projection, Riesz Representation Theorem.
Operators on Hilbert spaces: Adjoint of an operator, Normal, unitary, self-adjoint operators, positive operators and their spectra, Spectral theorem for compact self-adjoint operators (without proof).
- J. B. Conway, A course in functional analysis, GTM (96), Springer, 2007
- B. V. Limaye, Functional Analysis, 3rd Ed., New Age International Publishers, 2014
- M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Academic Press, 1980
- G. Bachman and L. Narici, Functional Analysis, 2nd Ed., Dover Publication, 1998
- S. Kesavan, Functional Analysis, TRIM series, Hindustan Book Agency, 2009
- Yosida, K., Functional Analysis, 6th Ed., Springer-Verlag Berlin Heidelberg, 1995
- Rajendra Bhatia, Notes on Functional Analysis, Hindustan Book Agency, 2015
- M. T. Nair, Functional analysis: A first course, PHI-Learning, New Delhi, Fourth Print, 2014