Quasinormability and asymptotic normability of spaces of entire functions of bounded type. (Q1396084)

The authors study the quasinormability and the asymptotic normability for spaces of entire functions of bounded type defined on duals of Fréchet spaces. They show that, if \(E\) is a quasinormable Fréchet space such that either \(E\) is hilbertisable or its strong dual \(E'\) has an absolute basis, then \(\mathcal{H}_b(E')\) is quasinormable. For asymptotically normable Fréchet spaces \(E\), it is shown that \(\mathcal{H}_b(E')\) is asymptotically normable. More general results concerning quasinormability of spaces of entire functions of bounded type, apparently unknown to the authors, can be found in [\textit{A. Peris}, Ann. Acad. Sci. Fenn. Ser. A. I, Math. 19, 167--203 (1994; Zbl 0789.46006)].