Schwartz kernel theorem for the N-Torus

Don't listen to algebraists and use a basis!

Here the Fourier basis works. For temperate distributions in $\mathbb{R}^d$ which is in fact a bit more difficult, you can use the basis of Hermite functions (or even better the Meyer wavelet basis).

Then deduce the wanted kernel theorem from that for matrices.

Let $\mathscr{s}(\mathbb{N}^d)$ denote the space of multi-sequences indexed by multiindices $x=(x_{\alpha})_{\alpha\in\mathbb{N}^d}$ with faster than polynomial decay. Let $\mathscr{s}'(\mathbb{N}^d)$ that of multi-sequences $y=(y_{\alpha})_{\alpha\in\mathbb{N}^d}$ of at most polynomial growth. The needed kernel theorem is a bijective correspondence between continuous linear maps $L: \mathscr{s}(\mathbb{N}^d)\rightarrow \mathscr{s}'(\mathbb{N}^d)$ and matrices $A=(A_{\alpha,\beta})_{\alpha,\beta\in\mathbb{N}^d}\in \mathscr{s}'(\mathbb{N}^{2d})$. It's easy to prove by hand.