ca.analysis and odes - What do we know about the space of finite order distributions ?

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

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