28393 - Functional Analysis 1

Course contents

Nets and filters. Topologies. Topological vector spaces (tvs). Hausdorff and locally convex tvs. Linear and continuous mappings. Completeness and completion of a tvs. Compactness in metric spaces and in tvs. Application to the proof of the theorem of Tychonov. Ascoli-Arzelà theorem. Hausdorff finite dimensional tvs. Bounded subsets of a tvs. Normability of a tvs. Important examples of normed spaces. Fréchet spaces with examples. Montel spaces. Hilbert spaces. Orthonormal systems with application to Fourier series. The Baire property. Open mapping and closed graph theorem for some applications (Hoermander's hypoellipticity theorem). Inductive limit topologies. Quotient spaces. LF spaces. Examples (space of polynomials, C^\infty functions with compact support). Approximation and density. Barreled spaces and Banach-Steinhaus theorem. Application to Fourier series. Geometric and analytic versions of the Hahn-Banach theorem. Theorems of separation. Duality. Polar topologies. Theorem of Banach-Alaoglu. Theorem of Mackey. Strong topology. REflexive spaces. Examples. Eberlein-Shmulyan theorem. Reflexivity of L^p spaces. Distributions. 

Readings/Bibliography

F. Treves, "Topological vector spaces, distributions and kernels",

Horvath, "Topological vector spaces and distributions",

Teaching methods

Lessons at the blackboard

Assessment methods

Oral examination.

Office hours

See the website of Davide Guidetti