Course Outline
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 ℓp, Lp.
- 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).
Last modified: Monday, July 31, 2017, 2:37 PM