Meromorphic functions with values in a Fréchet space and linear topological invariant (DN) (Q1808888)

Let \(X\) be a subset of \(\mathbb{C}^n\) and \(E\) a sequentially complete locally convex space. The spaces of meromorphic and weakly meromorphic functions on \(X\) with values in \(E\) are denoted by \(M(X,E)\) and \(M_w(X,E)\), respectively. In \textit{Le Mau Hai, Nguyen Van Khue} and \textit{Nguyen Thu Nga} [Colloq. Math. 64, Fasc. 1, 65-70 (1993; Zbl 0824.46052)] it is proved that \(M(X,E)=M_w(X,E)\) holds if \(E\) is a Banbch space and \(X\) is either open or compact. The authors show the following main results: Theorem 1. Let \(E\) be a Frechet space. Then \(E\) has a continuous norm iff \(M(X,E)=M_w(X,E)\) for every open subset \(X\) of \(\mathbb{C}^n\). Theorem 2. Let \(E\) be a Fréchet space. Then \(E\) has the property (DN) iff \(M(X,E)=M_w(X,E)\) for every \(\widetilde L\)-regular compact subset \(X\) of \(\mathbb{C}^n\). In Theorem 3 the authors give conditions in order to conclude the existence of an extension of real-analytic \(E\)-valued functions from the existence of weak extensions.