For U open in a locally convex space E it is shown in [31] that there is a complete locally convex space G(U) such that $G{\left(U\right)}_i^{\text{'}}=(\mathscr{H}\left(U\right),{\tau }_{\delta })$. Here, we assume U is balanced open in a Fréchet space and give necessary and sufficient conditions for G(U) to be Montel and reflexive. These results give an insight into the relationship between the ${\tau }_0$ and ${\tau }_{\omega }$ topologies on ℋ (U).