Literature recommendation on PDE

Question

I am looking for Partial Differential Equation (PDE) books that are primarily written for mathematicians, namely with an emphasis on proofs of (existence) general solutions to PDE's. To be more specific:

Ideally I'm most interested in a book that starts slowly with discussions on fourier transform in infinite dimensional systems, showing how it is defined for $L^1$ functions and how it can be extended to $L^2$ functions using Schwartz functions.

So in short if you know of such books, starting first by introductory concepts (fourier transform, convolutions, measure theory...) then building up to show how fourier analysis can be applied to PDE's, then please do suggest them here. (I said 'ideally' as I do not expect that books with such characteristics necessarily exist, so even if you know of references close to the description, feel free to mention them.)

Answer 1

Partial differential equations by Evans is kind of a canonical answer to such questions.

Answer 2

You could also look at Taylor's book series "Partial differential equations".