University of Illinois at Urbana-Champaign

Math 541 (Measure Theory and Functional Analysis) Spring 2007

Course Description

This course is a continuation of Math 540. It comes in three parts. In Part I we develop the theory of measure and integration in abstract spaces and then in greater depth in spaces with additional structure (topological or metric spaces). Part II is an introduction to the basics of functional analysis (Hilbert and Banach spaces, Hahn--Banach Theorem, Open Mapping and Closed Graph Theorems, Principle of Uniform Boundedness, weak topologies and Alaoglu's theorem). An important source of examples will be various function spaces defined over abstract measure spaces. In Part III we will give a crash course in the theory of Sobolev spaces and distributions.


Tentative Syllabus
  1. Measure and integration, Lp spaces, Lebesgue-Radon-Nikodym theorem, outer measure, constructions of measures
  2. Measure spaces with additional structure: measure and topology, measure and metric, Lebesgue differentiation theorem, Hausdorff measure
  3. Hilbert and Banach spaces, linear functionals, dual spaces, Riesz Representation Theorem, examples, Hahn-Banach theorem, reflexivity
  4. Topological vector spaces, weak topologies, Alaoglu theorem, applications
  5. Operator theory: Principle of Uniform Boundedness, Closed Graph and Open Mapping theorems, category
  6. Distributions and Sobolev spaces, fundamental solutions of linear PDE