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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: 1,523
Registered: 1/8/12
Re: countable set of closed subspaces in separable Hilbert space

Posted: Nov 11, 2012 12:56 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?

> 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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