An Introduction to Functional Analysis

Functional analysis is the branch of mathematics dealing with spaces offunctions. It is a valuable tool in theoretical mathematics as well asengineering. It is at the very core of numerical simulation.

In this class, I will explain the concepts of convergence and talk abouttopology. You will understand the difference between strong convergenceand weak convergence. You will also see how these two concepts can be used.

You will learn about different types of spaces including metric spaces,Banach Spaces, Hilbert Spaces and what property can be expected. You willsee beautiful lemmas and theorems such as Riesz and Lax-Milgram and I willalso describe Lp spaces, Sobolev spaces and provide a few details aboutPDEs, or Partial Differential Equations.

Week 1: Topology; continuity and convergence of a sequence in a topological space.
Week 2: Metric and normed spaces; completeness
Week 3: Banach spaces; linear continuous functions; weak topology
Week 4: Hilbert spaces; The Riesz representation theorem
Week 5: The Lax-Milgram Lemma
Week 6: Properties of the Lp spaces
Week 7: Distributions and Sobolev Spaces
Week 8: Application: simulating a membrane