### Math 541-Functional Analaysis

### Time/Location: MWF, 2-2.50 pm, Zoom

### Instructor: Marius Junge,

### Office hour: TBD

**Text: Conway: Functional Analysis
**

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

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