Time:  MWF 11-12 pm

Location:  Altgeld Hall 345

Instructor: Marius Junge            Office hours:  to be announced

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.
                            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
 

Course discription:

This course is intended as a
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, Hahn-Banach Theorem, Application to compact operators
II:  Properties of the integral, Outer measure, Fatou's lemma and convergence theorems, Fubini, Tonelli, Hilbert spaces, Radon Nikodym, Differentiation theorem
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
 

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