On dense subspaces of products of semibornological topological linear spaces (Q1057449)

It is proved the following theorem: Let \(E_ I\) be the topological product of an uncountable family \((E_ i)_{i\in I}\) of nonzero semibornological spaces. Let \(H_ I\) be its subspace formed by vectors which have at most countable many nonzero components, endowed with the induced topology. Then every linear subspace G of \(E_ I\) such that \(\oplus_{i\in I}E_ i\subset G\subset H_ I\) is semibornological, but no linear subspace G of \(E_ I\) such that \(H_ I\subset G\) and \(0<\dim (G/H_ I)<\infty\) is semibornological. (A topological vector space E is semibornological if every bounded linear functional over E is continuous.)