There is a standard way of endowing the Schwartz space $\mathcal{S}(\mathbb{R}^d)$ on $\mathbb{R}^d$ with a topology via semi-norms, turning it into a Fréchet space.

Now let $F$ be a non-archimedean local field. By definition, the Schwartz space $\mathcal{S}(F)$ is the space of all locally constant, compactly supported functions $F \rightarrow \mathbb{C}$. Is there a standard (canonical, natural) topology on $\mathcal{S}(F)$?

My thoughts: I remember vaguely from a conversation with my advisor, that one does this as follows: Let $A \subset F$ be the ring of integers of $F$ and let $\varpi \in A$ be a uniformizer. For integers $N \geq 1$ consider the subspace $$ \mathcal{S}(F)_N := \{f \in \mathcal{S}(F)\, :\, \text{supp}{f} \subset \varpi^{-N}A\quad , \quad f(x+ \varpi^{N}a) = f(x) \quad \forall (x,a) \in F \times A \} $$

Each of these spaces can be shown to be finite dimensional and as such has a natural topology. Moreover, we have $\mathcal{S}(F)_N = \bigcup_{N \geq 1}{\mathcal{S}(F)_N}$. Let $\iota_N : \mathcal{S}(F)_N \rightarrow \mathcal{S}(F)$ be the inclusion map. Does one endow $\mathcal{S}(F)$ with the topology, where a subset $U \subset \mathcal{S}(F)$ is open, if and only if $\iota_N^{-1}(U) \subset \mathcal{S}(F)_N$ is open for all $N$? In other words the strongest topology making all the inclusions $\rho_N$ continuous. This would have the effect that every linear functional $\mathcal{S}(F) \rightarrow \mathbb{C}$ is continuous (because all restriction to the $\mathcal{S}(F)_N$ are). Is that correct?

Now let $F$ be a global field. How does one topologize the Schwartz space $\mathcal{S}(\mathbb{A}_F)$ over its adele ring? My thoughts:

By definition, one has for each finite set $S$ of places of $F$ containing the infinite places, a linear map

$$T_S:= \bigotimes_{v \in S}{\mathcal{S}(F_v)} \longrightarrow \mathcal{S}(\mathbb{A}_F)$$

Also, if $S \subset S'$ are sets as above, then there is a linear map $T_S \rightarrow T_{S'}$ given informally "by filling up" the places $v \in S' - S$ with indicator functions of the local ring of integers of $F_v$. It seems to me that the family of spaces $\{T_S\}_S$ together with the family of maps $\{T_S \rightarrow T_{S}'\}$ forms a directed system of complex vector spaces and so $\mathcal{S}(\mathbb{A}_{F})$ (plus the maps $T_S$ into it) is a candidate for a colimit of that system (as a vector space).

Can one give the spaces $T_S$ a natural topology? Can one then use the topologies on the $T_S$ and the above observation to topologize $\mathcal{S}(\mathbb{A}_F)$?

  • $\begingroup$ Did you check it works well with this p.7-8 ? $\endgroup$ – reuns Nov 28 '17 at 22:56
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    $\begingroup$ Yes, as you essentially observe, Schwartz spaces on p-adic spaces are colimits of finite-dimensional topological vector spaces, so have a unique (locally convex) topology. Similarly, yes, the adelic Schwartz space is a (strict) colimit, which determines its natural topology. $\endgroup$ – paul garrett Nov 28 '17 at 23:03
  • $\begingroup$ Cool! But what about the topologies on the finite tensor products (the ones I denoted T_S)? $\endgroup$ – m.s Nov 28 '17 at 23:09

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