The jump from spherical harmonics to eigenfunctions on a general mesh, and the specific example mesh chosen, might be the finest mathematical joke I've seen this decade.
>If you’re alarmed by the fact that the set of all real functions does not form a HILBERT SPACE, you’re probably not in the target audience of this post."
Video: https://youtu.be/q8gng_2gn70?t=8m3s
Thanks to
> Dr. von Neumann, ich möchte gerne wissen, was ist denn eigentlich ein Hilbertscher Raum? (Dr. von Neumann, I'd would really like to know, just what exactly is a Hilbert space?)
Asked to John von Neumann to David Hilbert at a lecture.
https://ncatlab.org/nlab/show/Hilbert+space#fn:1
I'd like to add, as a physicist by training, that anything can be a Hilbert space if you wish hard enough. You can even use results about countable vector spaces if you need them!