Hi,

(Question updated)

My question is about the space of distributions of finite order $\mathcal{D}'_F$ (say on $\mathbb{R}^n$). What do we know about it ?

From in the information I gathered, it seems that the natural topology on $\mathcal{D}'_F$ is the inductive limit topology of the spaces $(\mathcal{D}'^m)$ of distributions of order $m$, or equivalently, the dual topology of $\mathcal{D}_F$ [ this space being $\mathcal{D}$ as a set, but with the coarser topology of the projective limit $(\mathcal{D}^m)$ ($C^m$ functions with compact support, this is an inductive limit of Fréchet spaces with obvious semi-norms). Note that $\mathcal{D}_F$ is strictly coarser than $\mathcal{D}$ (and strictly finer than the $\mathcal{S}$), and that $\mathcal{D}'_F$ is strictly finer than $\mathcal{D}'$ (and strictly coarser than $\mathcal{S'}$).

So, the question is: what do we now about this topology on $\mathcal{D}_F$, and its strong dual $\mathcal{D}_F'$ ? It is clearly not Frechet, but is it complete ? Montel? Barrelled ? Nuclear ? Reflexive ? More generally, do we have most of the nice properties of $\mathcal{D}'$ for $\mathcal{D}_F'$ ?

Thanks