M604 - Analysis II

Course No: 
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.
Reference Books: 
  1. D. S. Bridges, Foundations of Real and Abstract Analysis, GTM series, Springer Verlag 1997.
  2. G. B. Foland, Real Analysis: Modern Techniques and Their Applications (2nd ed.), Wiley-Interscience/John Wiley Sons, Inc., 1999.
  3. G. K. Pederson, Analysis NOW, GTM series, Springer-Verlag, 1991. 
  4. W. Rudin, Real and Complex Analysis, TMH Edition, Second Edition, New-York, 1962.
  5. W. Rudin, Functional Analysis, TMH Edition, 1974.
  6. K. Yosida, Functional Analysis, Springer-Verlag 1968.

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