FIRST SEMESTER
Special Topics in Functional Analysis (CM513)

CM513: Special Topics in Functional Analysis


Prof. Constantin P. Niculescu

 

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. Schauderís fixed point theorem. Kakutaniís 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.