Lecture 1, September 6, 2007

In the first part, we give a 45 minute introduction to the field, indicating that functional analysis is a blend of linear algebra and topology (in our case: the theory of metric spaces). It is the theory of topological vector spaces and (continuous linear) maps between them. A historically important theorem on Sturm-Liouville problems is formulated, claiming the completeness of the eigenfunctions of rather general differential operators. The quest for such results has stimulated the development of the functional analytic approach, and we intend to understand this theorem by the end of the course.

In the second part we set out on the preliminaries: linear algebra and metric spaces. We skip the linear algebra part (read R&Y 1.1 if you want, but there is nothing new here), so the second part was entirely devoted to metric spaces. Basic notions are recalled at a quick pace and two new results were stated without proof: the Stone-Weierstrass theorem for compact metric spaces (R&Y states only a particular case) and the Baire category theorem on complete metric spaces. Read R&Y 1.2 but do not try to construct the proofs of all the statements yourself. Do the homework exercises in the first series instead: these should give you a good opportunity to work with the theory of metric spaces and continuous maps between them. If you can solve these, then your working knowledge of metric spaces is fine.