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Functional analysis is the branch of mathematics dealing with spaces of
functions. It is a valuable tool in theoretical mathematics as well as
engineering. It is at the very core of numerical simulation.
In this class, I will explain the concepts of convergence and talk about
topology. You will understand the difference between strong convergence
and 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 will
see beautiful lemmas and theorems such as Riesz and Lax-Milgram and I will
also describe Lp spaces, Sobolev spaces and provide a few details about
PDEs, 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