There are some well-known books on mathematical physics that are commonly used in undergraduate courses around the world as an introduction to mathematical methods in physics and/or applied sciences. My feeling about these mathematical physics books is that they are meant to be accessible to a large audience, i.e. physicists, engineers, mathematicians and so on. Thus, these books usually cover a lot of topics but most of them are poor in mathematical rigor. I'm looking for a book that cover these topics more rigorously, putting things in their context (functional analysis, linear algebra etc). As an example, I don't want to learn how to properly calculate the coefficients of a Fourier series but rather to learn that these series are expansions in terms of orthonormal basis in $L^{2}([0,1],\mathbb{R})$ and so on. What are some rigorous mathematical physics books?
Note: As I mentioned, I know that these courses on mathematical physics usually cover a lot of stuff and the best approach here would be to get a book for each subject. But I would like to know if there is some book which puts it all together, in the same way the classical books do, but in a more concrete and rigorous way.
