Some universality questions for separable topological vector spaces (Q1092364)

This is a study in universality of topological vector spaces. A topological vector space (tvs) E is said to be universal for a class \({\mathfrak L}\) of tvs's, if each member in \({\mathfrak L}\) is embedded in E with \(E\in {\mathfrak L}\). The author first gave some account of basic properties of classes of Suslin and Lusin. From a result known as the Hewitt- Marczewski-Pondiczery theorem, a universal separable tvs has been constructed and there are precisely c such spaces. Also, note that there is no universal member in certain Lusin classes (or some Suslin), as well as almost all classes of tvs having a basis of one kind or another. The last example gives a firm negative answer to the universality of classes with such bases. Finally, with the modified condition, card (set of bases)\(\leq c\), there is a universal tvs with a basis for this set.