Math 541-Functional Analaysis
Time/Location: MWF, 2-2.50 pm, Zoom
Instructor: Marius Junge,
Office hour: TBD
Text: Conway: Functional Analysis
Homework- (30%), Final (40%), Presentation (30%)
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.
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
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
1) Baire's Theorem for metric spaces
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
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
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