There is a thesis available online which covers this material---Xavier Mary, Sous-espaces hilbertiens (Rouen). Despite the title, it is in english.

13(1964). Sorry about the chopped up form of my answer. It's the software.

ordering which can be used to define notions of convergence---see his article in Jour. d'Analyse Math.

as a space (with structure) in its own right. I don't recall him using a topology but he does use an

i did Independent misunderstand your query. but you van embed all your Hilbert spaces into a large tvs,say the measurable functions and then you have a situation which has been studied by L. Schwartz who considers the family of all hilbertien subspaces of a gives tvs (that is all Hilbert spaces which embed continuously into it)

Sorry, you are correct, of course. I was confusing Čech completeness and topological completeness. However, both notions coincide for metrisable spaces since they are equivalent to the space being $G_\delta$ in a compactification in this case. This supplies a second