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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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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: countable set of closed subspaces in separable Hilbert space question
Posted: Nov 11, 2012 11:24 AM
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On Sat, 10 Nov 2012 20:46:42 -0800 (PST), "Daniel J. Greenhoe"
<dgreenhoe@yahoo.com> 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}.
>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?


I can't imagine what you mean by lim_{n->infty} X_n
other than the union.

???

>Pointers to good references are especially appreciated.
>Many thanks in advance,
>Dan





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