Completeness of the (LB)-spaces \({\mathcal V}C(X)\) (Q1263791)

Let \({\mathcal V}\) be a decreasing sequence of strictly positive continuous functions \(v_ n\) on a completely regular Hausdorff space X. If \(Cv_ n(X)\) denotes the Banach space of all continuous complex-valued functions f on X for which \(\| f\|_ n=\sup_{x\in X}v_ n(x)| f(x)| <\infty\), \(n=1,2,...\), \({\mathcal V}C(X)=ind_ n Cv_ n(X)\) is the ``weighted (locally convex) inductive limit of spaces of continuous functions with \({\mathcal O}\)-growth conditions''. We prove that (under our assumptions) \({\mathcal V}C(X)\) is always complete by showing directly that it is a closed topological subspace of the corresponding co-echelon space \(k_{\infty}({\mathcal V})\). At the end of the paper, we add some remarks on the case of general (i.e., strictly positive and upper semicontinuous, but not necessarily continuous) weights \(v_ n\).