Spaces of continuous functions \(C_p(X,E)\) as \((LM)\)-spaces (Q2574784)

A topological vector space (tvs) (resp.,\ a locally convex space (lcs)) is called an \((LM)_{tv}\)-space (resp.,\ an \((LM)\)-space) if it is the inductive limit in the category TVS (resp.,\ in the category LCS) of an increasing sequence of metrizable spaces. A tvs is called \(b\)-Baire-like if every bornivorous sequence of closed almost strings in it is topological. The authors prove the following two results: (1)~Let \(E\) be a \(b\)-Baire-like tvs and \(F\) a dense subspace. Then \(F\) is an \((LM)_{tv}\)-space iff \(F\) (equivalently \(E\)) is metrizable. Hence, an \((LM)_{tv}\)-space is metrizable iff its completion is a Baire space, and \(\mathbb R^X\) (\(X\) is a completely regular topological space) contains a proper dense subspace which is an \((LF)\)-space iff \(| X| =\aleph_0\). (2)~Let \(C_p(X,E)\) be the space of continuous functions from a completely regular topological space \(X\) into a metrizable tvs \(E\), endowed with the pointwise topology. Then the following assertions are equivalent: (a)~\(C_p(X,E)\) is an \((LM)_{tv}\)-space; (b)~\(C_p(X,\mathbb R)\) is an \((LM)\)-space; (c)~\(| X| =\aleph_0\); (d)~\(C_p(X,E)\) is metrizable.