Time: MWF 11-12 pm
Location: Altgeld Hall 345
Instructor: Marius Junge Office hours: Thursday 11-12 am, Altgeld 363
Extra exercises: Tuesday 5-6pm Altgeld 343
Course e-mail: math34402@math.uiuc.edu
Books: We will blend in material from different
sources:
Conway, John B. A course in functional analysis. Second edition.Grading: 25% HW (submission in pairs), 25 % midterm, 50% final (or other alternatives to be discussed in class)
Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990.
Folland, B.: Real Analysis: Modern Techniques and Their Applications;
Wiley and Son Inc, 1999.
Pedersen, K.: Analysis Now:
Graduate Texts in Mathematics 118, Springer Verlag New-York.
Diestel, J.; Uhl, J. J., Jr. Vector measures. Mathematical Surveys, No. 15.
American Mathematical Society, Providence, R.I., 1977
Pages 41-57
Course discription:
This course is intended as aPlan: (The links contain a sceleton of the material)
continuation of Math 441 and should provide a good background in analysis which is useful for a wide range of areas (real and complex analysis, PDE, ODE, probability and operator algebras).
We will start with basic notions in Banach spaces and then review and extend the notion
of integrable functions. Indeed, we will cover the Fubini theorem for arbitrary measure spaces and for functions with values in several dimensions. In the second part, we will apply these concepts and introduce the Fourier transforms, the Haussdorff-Young inequality and the representation of functions on the torus by Fourier coefficients. In the third part we will discuss Baire's category theorem and its applications (closed graph, open mapping and the uniform boundedness principle, convergence of Fourier series). If time
permits, we will conclude the course by discussing certain locally convex topologies coming from duality pairs.
I: Introduction
to Banach spaces (Conway 63-70), Hahn-Banach Theorem (Conway 73-77),
Application to compact operators (Conway 41)
II: Properties of the integral (Folland
43-49), Outer measure (Folland 19-40), Fatou's lemma and convergence theorems
(Folland 60-64), Fubini (Folland 64-69), Tonelli (Folland 64-69), Hilbert
spaces (Conway 1-23), Radon Nikodym (Pederson) , Differentiation theorem
(Folland 95 ff)
III: Some Fourier series and Fouries analaysis
including the Baire categroy theorem and the open mapping theorem
Detailed version of the course material
The completion of a normed space
Introduction to Banach spaces by Aleksander Jovicic
Compact operators by Martin Singleton
Homework