Math 541-Functional Analaysis


Time/Location: MWF, 2-2.50 pm,  Zoom 

Instructor: Marius Junge, 

https://illinois.zoom.us/j/89330242963?pwd=dTg0ZTkwOWUxMVgwaXNKa01EZVFKdz09

Office hour: TBD

Text:  Conway: Functional Analysis

Grading:  Homework- (30%),  Final  (40%),  Presentation (30%)

Grader: Haojian Li

Course email

Outline: The aim of this course is to outline the so called 5 highlights in functional analysis. Today functional analysis  should be considered a tool for many aspects of  analysis, such as harmonic and  Fourieranalysis, operator algebras, Banach space theory, and certain aspects of applied math including quantum information theory.  The theory  has become decievingly  simply and beautiful over time.
 
At the end of the course we will focus on applications of functional analysis, or if you whish, linear algebra in infinite dimension. Please inform the lecturer about your preferences.

Organization:  The courses will be live zoom meetings, the link will be sent to you by email. The zoom meetings will also be recorded, and made
                               avaialble after a couple of days. 
                               The course material will be in the Box folder (please ask for access),
                               Homework will be submitted to your folder hw-assignement with your name on          
 

Topics:

1)   Baire's Theorem for metric spaces

2)  Basic Banach spaces, Hahn-Banach Theorem, topological vector spaces, locally convex spaces, Hahn-Banach seperation theorems.

3) Abstract Banach space theory: weak and weak* topology, Alaoglu and Goldstine, Reflexivity, Uniform convexity.

4) Concrete Banach space duality: Clarkson inequalities, L_p spaces for measure spaces, martingale convergence theorem and uniform convexity,

5) open mapping theorem and closed graph theorem

6)  Riesz representation theorem and Krein-Milman.


7) Hilbert spaces and best aproximation, duality, compact operators, finite rank operators and duality.

8) Compact operators and duality  

9)  Spectral  theory of compact operators and Fredholm algernative.
     

10)  Further  applications
 

 


Old homework

HW1, HW2, HW3, HW4, HW5, HW6, HW7, HW8, HW9, HW10, HW11

List of presentation problems


Links and material

Book: Conway, John B.  A course in functional analysis.
Graduate Texts in Mathematics, 96. Springer-Verlag, New York,  1985. xiv+404 pp. ISBN: 0-387-96042-2

Additional material:

Minmax theorem by Borwein and Zhuang

Minimax theorem by David Pollard

Paper by Boas

Uniform convexity of L_p(X) by Mahlon Day

Riesz-Markov by Arveson

Courses webpage
Department of Mathematics