On representations of entire functions by Dirichlet series in infinite dimension (Q1373068)

Let \(\mathcal B (E)\) denote the family of all closed bounded balanced convex subsets of a locally convex space E. For \(B\in \mathcal B (E)\) let \(E(B)\) be the normed space spanned by \(B\) and \(H_b(E(B))\) be the Fréchet space of holomorphic functions on \(E(B)\) which are bounded on every bounded set in \(E(B)\). The main result in this paper is the following: Let \(E\) be a nuclear Fréchet space. Then for each \(K\in \mathcal B (E),\) there exists an increasing sequence \(\{ K_n\} \subset \mathcal B (E)\) with \(K_1=K \) and sequences \(\{ x_j^n \}\subset E(K_n)\) such that \(\sum_j \exp (-\| x_j^n \| _{K_n}) < \infty,\) for \(n\geq 1\). Every \(f\in \bigcup_n H_b([E(K_n)]^*)\) can be written in the form \(f(x^*)=\sum_j \xi_j\exp <x_j^n,x^*>\) for \(x^*\in [E(K_n)]^*\) with some \(n=n_f\) for which \(\sum_j| \xi_j | \exp (r\| x_j^n \| _{K_n}) < \infty\) for all \(r>0\).