Banach spaces: Review of Banach spaces, Hahn-Banach Theorem and its applications, Baire Category theorem and its applications like Closed graph theorem, Open mapping theorem.
Topological Vector spaces: Weak and weak* topologies, locally convex topological vector spaces.
Hilbert spaces: Review of Hilbert spaces and operator Theory, Compact operators, Schauder's theorem on the spectral theory of compact operators.
Banach algebras: Elementary properties, Resolvent and spectrum, Spectral radius formula, Ideals and homomorphisms, Gelfand transforms, Gelfand theorem for commutative Banach algebras.
- D. S. Bridges, Foundations of Real and Abstract Analysis, GTM series, Springer Verlag 1997.
- G. B. Foland, Real Analysis: Modern Techniques and Their Applications (2nd ed.), Wiley-Interscience/John Wiley Sons, Inc., 1999.
- G. K. Pederson, Analysis NOW, GTM series, Springer-Verlag, 1991.
- W. Rudin, Real and Complex Analysis, TMH Edition, Second Edition, New-York, 1962.
- W. Rudin, Functional Analysis, TMH Edition, 1974.
- K. Yosida, Functional Analysis, Springer-Verlag 1968.