I am looking for references (books, lecture notes etc) on mathematical classical field theory. By that, I mean classical field theory under a rigorous point of view. However, I am more interested in the functional analysis approach. Let me explain a little more below.

The standard mathematical formalism to study classical mechanics is symplectic geometry. Under a mathematical point of view, it is more convenient (and more general) to consider manifolds instead of, say, $\mathbb{R}^{d}$. The underlying concepts of the theory, such as principle of stationary action, is also conveniently expressed in terms of these abstract tools. The same holds true when one passes to other classical theories such as electromagnetism and classical field theory.

However, I am looking for a functional analysis approach, which I consider to be more closely related to what physicists do, at least in a first approximation. In a regular course of classical field theory, the Klein-Gordon, Dirac and Electromagnetic fields are usually addressed. The principle of stationary action is also presented. These fields are usually element of the space of rapidly decreasing functions $\mathscr{S}(\mathbb{R}^{d})$, and they solve their respective differential equation. The use of functional derivatives in the principle of stationary action is, I believe, connected to the famous Schwartz Kernel Theorem from the theory of distributions. Hence, it is quite natural to expect that a rigorous presentation of these concepts under a functional analysis point of view must be available somewhere. However, I have not found any reference which covers the above and using this language I am proposing. Therefore I am looking for suggestions and I hope you can help me find some good ones!

In summary, what I expect to find:

• The formulation of the Klein-Gordon, Dirac and Electromgnetic fields as PDEs in a proper space of rapidly decreasing functions and the discussion of its solutions.
• A good discussion of functional derivatives (probably in locally convex spaces, or simply in $\mathscr{S}(\mathbb{R}^{d})$ and the use of Schwartz Kernel Theorem to explain the physicist notation of functional derivative.