In this note we prove that the space $[\mathcal{H}(O_E)]^\prime$ is asymptotically normable (resp. has property $(\underline{DN})$) if E is an asymptotically normable Frechet space (resp. $E$ is a Frechet $(\underline{DN})$-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 $\mathbf{C}^n$.