MTH402: Functional analysis

Course Outline

• Normed linear spaces, Banach spaces, examples and interesting dense subspaces.
• Continuous linear functionals, duals.
• Hahn-Banach theorem, separation theorems.
• Duals of classical spaces ℓ^p, L^p.
• Bounded linear operators, open mapping and closed graph theorems, uniform boundedness principle and applications, spectrum of an operator.
• Hilbert spaces, orthogonality and geometric structure, projections, Reisz representation theorem. Fourier series and L² theory.
• The Banach space B(H)
• Adjoint of an operator, self-adjoint, normal and unitary operators. spectral theorem for compact self-adjoint operators.

Additional Topics

• Weak and weak-* topologies, Banach-Alaoglu theorem.
• Spectral theorem for general self-adjoint and normal operators. Reisz representation theorem, dual of C₀(X) for X a locally compact space, Gelfand theory.
• Unbounded operators: definition and examples.

Recommended Reading

• J. B. Conway, A course in Functional Analysis, Springer (Graduate Texts in Mathematics Vol. 96) (1990).
• G. F. Simmons, Introduction to Topology and Modern Analysis,Tata McGraw Hill (2004).
• B. Bollobas, Linear Analysis: An Introductory Course, Cambridge University Press, Cambridge (1999).
• B. V. Limaye, Functional Analysis, New Age International Publishers Limited, New Delhi (1996).