Systems of absolute representations and prolongations of entire functions in dual spaces of Fréchet-Montel (Q1373040)

For Fréchet spaces \(E\) and \(F\) denote by \(E'\) the strong dual of \(E\) and by \({\mathcal O}(E',F)\) the space of all holomorphic functions on \(E'\) with values in \(F\). It is shown that \(E\) is nuclear, if and only if for each Banach space \(F\) and each \(f\in{\mathcal O}(E', F)\) there exist sequences \((\xi_j)_{j\in\mathbb{N}}\) in \(F\) and \((x_j)_{j\in\mathbb{N}}\) in \(E\) satisfying \[ \sum_{j\in\mathbb{N}}\|\xi_j\| \exp(\sup\{|\langle z,x_j\rangle|: z\in L\})< \infty \] for each compact set \(L\) in \(E'\) such that \[ f(y)= \sum^\infty_{j= 1} \xi_j\exp(\langle y, x_j\rangle),\quad y\in E'. \] Also it is shown that this way of representing entire functions on (DFM)-spaces can be used to improve an extension result of \textit{P. J. Boland} [Trans. Am. Math. Soc. 209, 275-281 (1975; Zbl 0317.46036), see also \textit{R. Meise} and \textit{D. Vogt}, Proc. Am. Math. Soc. 92, 495-500 (1984; Zbl 0561.46024)].