28393 - Functional Analysis 1

Academic Year 2021/2022

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Mathematics (cod. 8010)

Learning outcomes

At the end of the course, the student should have acquired the basic competences of functional analysis and of the theory of bounded and linear operators. 

Course contents

Nets and filters. Topologies. Topological vector spaces (tvs), Haussdort tvs and locally convex tvs. Metrizable tps. Completeness and completion of a tvs. Compactness in metric spaces and tvs. Application to a proof of Tychonov theorem. The theorem of Ascoli-Arzelà. Haussdorf and finite dimensional tvs. Normability of a tvs. Important examples of normed spaces. Fréchet spaces with several examples. Montel spaces. Examples. Hilbert spaces. Orthonormal systems and application to Fourier series. The property of Baire. Theorems of the open mapping and the closed graph with some applications (in particular Hoermander's ipoellipticity theorem). Inductive limit topologies. Quotient topologies. LF spaces. Examples (spaces of polynomals, space of smooth functions with compact support). Results of approximation and density. Barrelled spaces and Banach-Steinhaus theorem. Application to the proof of the existence of a continuous, periodic function with divergent Fourier series. Geometric and analytic versione of the Hahn-Banach theorem. Separation theorems. Duality. Polar topologies. Banach-Alaoglu theorem. Mackey theorem. 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

Front lectures or (if necessary) on line. 

Assessment methods

Oral examination

Teaching tools

Blackboard

Office hours

See the website of Davide Guidetti