Functional analysis is a powerful, modern branch of mathematical analysis where topics that are familiar from the theory of real numbers (sequences, series, convergence, etc) are generalized to the infinite dimensional world of functions. This talk provides an introduction to this field for students who have mastered elementary real analysis, tying together both historical motivations and the modern viewpoint.
The first half of the talk focuses on the pre-history of the subject, looking at the mathematics of the geocentric model of the heavens developed by the Ptolemies and Ibn al-Shatir, among others. While the notion of epicycles very naturally leads to precise mathematical questions about sequences of functions and convergence, it took us many further centuries to fully understand what lies just beneath the surface of this idea.
In the second half of the talk we skip ahead these centuries and visit another time period where the same worries of sequences and convergence arose once more: with Fourier and his study of heat. Taking advantage of our modern perspective, we work to formalize the questions these historical figures may have asked, and then lay out the roadmap leading from here to the modern theory of Functional Analysis.
This series of talks was written to take place during the final week of Math 115, an undergraduate real analysis course I taught virtually at Stanford during the pandemic.
Here’s the combined slide deck from these lectures!