Bornologicity of certain spaces of bounded linear operators (Q2714303)

A locally convex space \(E\) carries a convex bornological structure (a collection of bounded sets) given by the associated Von Neumann bornology: for such \(E\), the space \(L(E,E)\) of bounded linear operators on \(E\) is usually endowed with the topology of uniform convergence on bounded sets yielding a locally convex structure. In general \(L(E,E)\) with this structure is not bornological. NEWLINENEWLINENEWLINEAn example of a nuclear Fréchet space \(E\) is given, for which \(L(E,E)\), with the structure given above, is not bornological. It has been shown that for \(E\) the subspace of the product \(R^I\) over an uncountable index set \(I\) formed by the sequences with countable support, this locally convex topology on the space \(L(E,E)\) is bornological.