In this note we prove that the space $[\mathcal{H}(O_E){]}^{\prime }$ is asymptotically normable (resp. has property $(\underset{―}{DN})$) if E is an asymptotically normable Frechet space (resp. $E$ is a Frechet $(\underset{―}{DN})$-space) having absolute basis. We also investigate these properties of the space $[\mathcal{H}(K){]}^{\prime }$ when $K$ is a balanced compact subset of an asymptotically normable Frechet-Hilbertisable space and $K$ is a polynomially convex conpact set in ${\mathbf{C}}^n$.