A note on the Baire and ultrabornological property of spaces \(C_ p(X,E)\) (Q1359095)

Let \(X\) be a completely regular Hausdorff topological space and \(E\) a Fréchet space, i.e. a metrizable and complete locally convex space. Let \(C_p(X,E)\) be the space of all continuous vector-valued functions on \(X\) into \(E\) endowed with the topology of pointwise convergence. It is proved that provided \(X\) is a Lindelöf \(P\)-space, then every sequentially closed linear subspace of \(C_p(X,E)\) is an ultrabornological Baire space but \(C_p(X,E)\) is the inductive limit of an increasing sequence of metrizable locally convex spaces iff \(X\) is countable.