28393 - Functional Analysis 1

Learning outcomes

At the end of the course the student has acquired the basic competences concerning linear functional analysis and the theory of linear continuous operators. He (she) is able to use the acquired knowledge to solve model problems connected with the theory of partial differential equations appearing in mathematical models of applied sciences. 

Course contents

Nets and filters. Topologies. Topological vector spaces (tvs). Haussdorf and locally convex tvss. Linear and continuous mappings. Metrizable tvss. Completeness and completion of a tvs. Compactness in metric spaces and in tvss. Application to the proof of Tychonov theorem. Ascoli-Arzelà theorem. Haussdorf tvss with finite dimension. Bounded subsets of a tvs. Normability of a tvs. Important examples of normed spaces. Fréchet spaces with several examples. Montel spaces. Examples. Hilbert spaces. Orthonormal systems with application to Fourier series. The Baire property. Theorems of the open mapping and the closed graph with some applications (in particular Hormander's ipoellipticity theorem). Inductive limit topologies. Quotient topologies. LF spaces. Examples (spaces of polynomials, spaces of smooth functions with compact support). Results of approximation and density. Barrelled spaces and Banach.Steinhaus theorem. Application to the existence of periodic continuous functions with divergent Fourier series. Geometric and analytic versions of the Hahn-Banach theorem. Theorems of separation. Duality. Polar topologies. Banach-Alaoglu theorem. Mackey theorem. Strong topology. Reflexive spaces. Examples. Eberlein-Shmulyan theorem. Reflexivity of L^p spaces. 

Readings/Bibliography

Notes of the teacher

Teaching methods

"Front" lectures.

Assessment methods

Oral examination with solution of a proposed exercise. 

Office hours

See the website of Davide Guidetti