So, I am trying to find a reference for a proof of Schwartz Kernel Theorem for $2\pi$-Periodic functions, i.e,
given a continuous and linear operator $A:\mathcal{D}(\mathbb{T}^N)\to \mathcal{D}'(\mathbb{T}^N)$ there exists $k\in\mathcal{D}'(\mathbb{T}^N\times \mathbb{T}^N)$ a unique periodic distribution such that
$$\langle{A\varphi,\psi}\rangle=\langle k,\psi\otimes \varphi\rangle, \forall \varphi,\psi\in \mathcal{D}(\mathbb{T}^N).$$
I am fine with it being a Corollary of Schwartz Kernel Theorem in $\mathbb{R}^N$ (it is obvious?). I know there is a proof for smooth manifolds, but maybe for $\mathbb{T}^N$ there is simpler one?
Best Answer
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.