MATH 541-Functional Analysis

**Time:** MWF 12:00-12:50pm

**Location: ** 347 Altgeld Hall

Course description :

Functional Analysis, Math 541, is one of the jewel among the analysis courses in our graduate school. In a nutshell one can say that functional analysis is linear algebra in infinite dimension, with just the right amount of topology to make it managable. The area of functional analysis was born out of different fields such as Banach spaces, operator algebra and ODE's discovering that a few principlies are useful in all of them. I invite the participants in this course to find out what is their favorite application.

Topics:

I: Review of Banach space properties ( see here)

II: LCTVS : Topologies, toplogical vector space, locally convex topologicall spaces and semi-norms, examples

III: Hahn Banach Theorem: Extension, Seperation, strict separation, applications.

IV: Krein-Milman and Caratheordory

V: Barire Category theorem and open mapping, closed graph, and uniform boundedness principle, application to Fourier series, closed opertators.

VI: Uniform convexity and applications to duality and Radon-Nikodym property, and other approximation tricks.

VII: Spectral Theory for compact operators

Comment: Weak toplogies will be disccussed in II and III, and IV. I will watch out for some applications in quantum information theory.

Books:

John Conway: A course in Functional Analysis, second eddition

Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo Graduate Texts in Mathematics 96

Grading:

Homework (50%)Final/Presentation (50%)

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