Nonseparable closed vector subspaces of separable topological vector spaces

Abstract

In 1983 P. Domański investigated the question: For which separable topological vector spaces E, does the separable space have a nonseparable closed vector subspace, where $\hbox {c}$ is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line $\mathbb M$ the space of all continuous real-valued functions on $\mathbb M$ endowed with the pointwise convergence topology, $C_p(\mathbb M)$ contains a nonseparable closed vector subspace while $C_p(\mathbb M)$ is separable.