What are Hilbert and Sobolev spaces?

There were some questions regarding some of the mathematical preliminaries, specifically regarding Hilbert and Sobolev spaces. So I thought I’d start a thread here where people can ask questions and provide additional reading material / insight.

Here’s a “short” textbook from Prof. Juha Kinnunen at Aalto Universitry in Finland:

Sobolev Spaces

Particularly I thought this excerpt from the very first chapter helped put things into context:

The Sobolev space is a vector space of functions that have weak derivatives. Motivation for studying these spaces is that solutions of partial differential equations, when they exist, belong naturally to Sobolev spaces

And here’s a chapter from another textbook with more discussion, including Hilbert spaces