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Topic: countable set of closed subspaces in separable Hilbert space question
Replies: 14   Last Post: Nov 13, 2012 10:29 PM

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William Elliot

Posts: 1,530
Registered: 1/8/12
Re: countable set of closed subspaces in separable Hilbert space
question

Posted: Nov 11, 2012 4:09 AM
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> > 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?




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