
Re: countable set of closed subspaces in separable Hilbert space question
Posted:
Nov 11, 2012 4:09 AM


> > On Sat, 10 Nov 2012, Daniel J. Greenhoe wrote: > > > > > Let H be a separable Hilbert space. > > > > > Let (X_n) be a sequence of nested subspaces in H such that > > > > > X_n subset X_{n+1}, X_n not= X_{n+1}. > > > > Vectorial subspaces or topological subspaces?
Ok, vectoral subspaces. What about closure? Do you mean the topological closure or some algebra construction?
As it seems lim(n>oo) X_n = \/_n X_n,, statements 1 and 2 appear equivalent. How are you defining lim(n>oo) X_n?
> > > What is the relationship between the following two conditions in H? > > > > > 1. closure{ Union X_n } = H > > > > > 2. closure{ lim_{n>infty} X_n } = H
> > > Does one imply the other? Are they equivalent?
> > Is there a difference between lim(n>oo) X_n and \/_n X_n? > > > > Closure in the topological sense?

