Course Descriptions

16:640:507 - Functional Analysis I

Sagun Chanillo

Course Description:

This will be a basic Functional Analysis course covering the three major theorems, the Hahn-Banach theorem, Uniform boundedness principle and the Open mapping-Closed Graph theorem. We shall also do Fredholm theory as it is useful to people doing PDE and also the Spectral theory of self-adjoint and bounded operators. The course will emphasize applications of Functional Analysis to PDE via illustrations in the use of Sobolev spaces.

Text: 

Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, Universitext, Springer-Verlag

Prerequisites: 

501, 502

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Fall 2020 Semester:

Hector Sussmann

Course Description:

The basic structures of linear functional analysis: Banach spaces, Hilbert spaces, locally convex topological vector spaces. Examples of spaces of functions and function-like objects: Lebesgue spaces, Sobolev spaces, spaces of holomorphic functions, spaces of distributions. Bounded linear operators. The standard classical theorems: Hahn-Banach, open mapping, closed graph, Banach-Steinhaus, Bourbaki-Alaoglu, nonemptyness of the spectrum, elementary properties of the resolvent. Spectral theory of compact operators. Applications to eigenfunction expansions. The Schauder fixed point theorem. Extreme points of convex sets in locally convex spaces and the Krein-Milman theorem. Elementary facts about Hilbert spaces.

Text: 

"Functional Analysis,  Sobolev Spaces and Partial Differential Equations" 

(Universitext) by Haim Brezis,  Springer, ISBN-13: 978-0387709130,   ISBN-10: 0387709134 

Prerequisites: 

640:502 or equivalent


Schedule of Sections:

Previous Semesters: