Spaces $C^\infty(\mathbb T^n\times \mathbb R^n)$, $C^\infty_0(\mathbb T^n\times \mathbb R^n)$ and $\mathscr{S}(\mathbb T^n\times \mathbb R^n)$? [closed]

Question

Is there any characterization of the space $C^\infty(\mathbb T^n\times \mathbb R^n)$ that I can take as a definition of it?

I assume it would be something like this:

$$C^\infty(\mathbb T^n\times \mathbb R^n):=\{f\in C^\infty(\mathbb R^n\times \mathbb R^n): f(x+\xi, y)=f(x, y)\ \forall \xi\in\mathbb Z^n\},$$ Is it ok? And what about $C^\infty_0(\mathbb T^n\times \mathbb R^n)$?

Secondly, what would be a reasonable definition of (an analogue of) the Schwartz space on $\mathbb T^n\times \mathbb R^n$? Maybe:

$$\mathscr{S}(\mathbb T^n\times \mathbb R^n)=\{f\in C^\infty(\mathbb T^n\times \mathbb R^n): f(t, \cdot)\in \mathscr{S}(\mathbb R^n)\ \forall t\in \mathbb T^n\}?$$

If those spaces are treated somewhere I would appreciate references.

Thanks.

Answer 1

On any abelian Lie group $G = \mathbf R^n\times\mathbf T^p\times\mathbf Z^q\times \Phi$ ($\Phi$ finite), one defines as usual $C_0^\infty(G)=\{$smooth functions on $G$ with compact support$\}$. (The notion makes sense on any manifold.)

As for $\mathscr S(G)$, Bruhat (1956, p.138) defines it as the space of all smooth $f$ on $G$ such that, for every polynomial $P$ on $\mathbf R^n\times\mathbf Z^q$ (viewed as a function on $G$) and every invariant differential operator $D$ on $G$, the function $P(g)Df(g)$ is bounded on $G$. (It is topologized by the resulting seminorms $\sup_g|P(g)Df(g)|$.)