Date: Nov 13, 2012 4:25 PM
Author: Achimota
Subject: Re: countable set of closed subspaces in separable Hilbert space question

On Sunday, November 11, 2012 12:46:42 PM UTC+8, 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}.
> 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?

Thank you to everyone who helped me with this question. It seems the likely conclusion is that they are essentially equivalent by definition, as suggested from the beginning by David and later also by Seymour. That is,
lim_{n->infty} X_n := closure{ Union X_n }
Many thanks to William for helping me see this.