1. Operators on Hilbert
spaces. The adjoint of an operator. Compact operators. Diagonalization of compact
self-adjoint operators. Applications to Sturm-Liouville problems. Spectral
theorem and functional calculus for normal operators. Unital equivalence for
normal compact operators.
2. Locally convex spaces. Metrizable and normable spaces.
Geometric consequences of the Hahn-Banach theorem. The dual space of a locally
convex space.
3. Weak topologies. The theorem of Alaoglu-Bourbaki.
Reflexive spaces. The Krein-Milman theorem. Schauder’s fixed point theorem.
Kakutani’s fixed point theorem. Applications.
4. Fourier transform. Sobolev spaces and Distributions.
Applications to PDE.
5. Unbounded operators. Symmetric and self-adjoint
operators. The Cayley transform. Unbounded normal operators and the spectral
theorem. Semigroups of operators. The Stone and the von Neumann theorems.
Bibliography
1. H. W. Alt, Lineare Funktionalanalysis, Springer-Lehrbuch,
Berlin, 1992.
2. R. Cristescu, Functional Analysis, Ed.
Didactica si Pedagogica, Bucharest, 1972 (Romanian).
3. J. B. Conway, A Course in Functional Analysis,
2nd ed., Springer-Verlag, Berlin, 1997.
4. W. Rudin, Analyse fonctionelle,
Ed. Ediscience International, 1995
5. K. Yosida, Functional Analysis, 5th
ed., Springer-Verlag, Berlin, 1995.