First holomorphic cohomology group and linear topological properties (Q1573542)

Let \(E\) and \(F\) be nuclear Fréchet spaces, \(E\) having property \((DN)\), \(F\) having property \((\Omega)\). Denote by \({\mathcal O}^{E^*}_{F^*}\) the sheaf of holomorphic functions on \(F^*\) with values in \(E^*\). Extending a result of \textit{J. F. Colombeau} and \textit{B. Perrot} [Bull. Soc. Math. Fr. 110, 15-26 (1982; Zbl 0493.46041)] for scalar functions the author claims that \(H^1(F^*,{\mathcal O}^{E^*}_{F^*})= 0\). However, at the beginning of the proof of Lemma 2 the reviewer could not see why one can find a polydisc \(D\) with the required properties.