Open mappings and the lack of full completeness of $\mathcal {D}’(\Omega )$

Consider a linear map $T$ of one locally convex linear space into another which is densely defined and has a closed graph. We characterise the property that $T$ is an open map in terms of two properties of its adjoint map ${T^{\ast }}$. These results are used to show that if $\Omega$ is an open subset of ${R_n}$ for which there is a linear constant coefficient differential operator $P$ such that $\Omega$ is $P$-convex but not strongly $P$-convex, then (i) $\mathcal {D}’(\Omega )$ is not fully complete, (ii) the range of the adjoint map $^tP$ is closed but not bornological.