Holomorphic functions on Fréchet-Montel spaces
Let \(E\) denote a locally convex space over the field of complex numbers \(\mathbb{C}\) and let \(U\) denote an open subset of \(E\). We let \(H(U)\) denote the space of \(\mathbb{C}\)-valued holomorphic functions on \(U\) and \(\tau_0\) is the compact open topology on \(H(U)\). A semi-norm \(p\) on \(H(U)\) is said to be ported by the compact subset \(K\) of \(U\) if for each open set \(V\), \(K\subset V\subset U\), there exists \(c(V)>0\) such that \(p(f)\leq c(V)|f|_V\) for all \(f\) in \(H(U)\). The locally convex topology on \(H(U)\) generated by the semi-norms ported by compact subsets of \(U\) is denoted by \(\tau_w\). Clearly, \(\tau_0\leq \tau_w\) and a good deal of research has been devoted to finding conditions under which we have \(\tau_0= \tau_w\). In this article we consider this problem for certain Fréchet spaces. In this situation the equality \(\tau_0= \tau_w\) on \(H(U)\) implies that \(U\) is an open subset of a Fréchet-Montel space and so, at the outset, we are confined to such spaces. \dots Our main result is the following Theorem 1. If \(E\) is a Fréchet-Montel space with an unconditional basis of type \((T)\) and \(U\) is a balanced domain in \(E\) then \(\tau_0= \tau_w\) on \(H(U)\).
scientific article; zbMATH DE number 62789
scientific article