Approximation of holomorphic functions of infinitely many variables. II. (Q1977474)

Let \(X\) be a Banach space. An unconditional basis of \(X\) is a subset \((e_\gamma)_{\gamma \in \Gamma}\) with the property that any \(x \in X\) can be represented uniquely as \(x = \sum_{\gamma \in \Gamma} x(\gamma) e_\gamma\). Typical examples of spaces with an unconditional basis are \(\ell^p(\Gamma)\), \(1 \leq p < \infty\). The main result of the paper under review is a solution of the following approximation problem for a Banach space \(X\) with a countable unconditional basis or a space \(\ell^p(\Gamma)\), \(\Gamma\) arbitrary: Let \(V\) be a sequentially complete locally convex space and \(f : B(R) \to V\) a holomorphic function on the open ball \(B(R)\) of radius \(R\). Then for each \(r \in ]0,R[\), \(\epsilon > 0\) and each continuous seminorm \(\psi\) on \(V\), there exists a holomorphic function \(h : X \to V\) such that \(\psi(f - h) < \varepsilon\) on \(B(r)\). As a consequence of this result, it follows from results in [\textit{L. Lempert}, Invent. Math. 142, No. 3, 579-603 (2000)] that if \(V\) is a Fréchet space and \({\mathcal V}\) the sheaf of germs of \(V\)-valued holomorphic functions on an open pseudoconvex subset \(\Omega \subseteq X\), then \(H^q(\Omega, {\mathcal V}) = 0\) for \(q \geq 1\). [For part I see Ann. Inst. Fourier 49, No.~4, 1293-1304 (1999; Zbl 0944.46046)].