Fall 2023
Fioralba Cakoni
Course Description:
This course will cover fundamental topics of linear functional analysis. Topics include (but not limited to): The Hahn-Banach Theorems, The Uniform Boundedness Principle and Closed Graph Theorems, Weak Topologies, Reflexive Spaces, Separable Spaces, Uniform Convexity, Hilbert Spaces, Compact Operators, Spectral Theory for Compact and Selfadjoint Operators.
Text:
Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haïm Brézis
Prerequisites:
Math 501, Math 502
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Fall 2022
Maxime Van De Moortel
Course Description:
The course will cover the fundamentals of Functional Analysis and will include the following topics:
I. Abstract theory of Banach spaces and applications to Lebesgue/Sobolev spaces.
1) The Hahn-Bach theorem and connections between convexity and continuity
2) Topology and duality of Banach spaces.
3) Lebesgue spaces: duality, compactness, and density results.
4) Sobolev spaces, Sobolev embedding, and Poincaré inequalities.
II. The theory of unbounded operators and applications to Partial Differential Equations.
1) The uniform boundedness principle and an introduction to unbounded operators theory.
2) Existence of solutions of PDEs using the Hahn-Banach theorem and uniform boundedness principle.
3) Compact operators and the Fredholm alternative.
4) Hilbert spaces: Lax-Milgram Theorem and applications to elliptic PDEs. Introduction to Spectral Theory (time-permitting).
Time-permitting, we will also discuss semi-groups of evolution and the Hille-Yosida theory. The course content will be essential to all students interested in PDEs and, more generally, Analysis and Applied Mathematics. We will discuss many illustrating examples to articulate the important connection between abstract theory and concrete applications. The grade will consist of a midterm oral presentation (10-15 mn) on a topic related to Functional Analysis of the student’s choice (suggestions of topics will be provided) and a take-home final exam.
Grade: Midterm oral presentation (40%) and take-home final written Exam (60%).
Text:
The course will primarily be based on Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haïm Brézis, with occasional deviations from the textbook.
Prerequisites:
Prior exposure to measure theory and elementary topology (e.g. in 501/502) is essential.
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Fall 2021
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:
