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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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Achimota

Posts: 254
Registered: 4/30/07
countable set of closed subspaces in separable Hilbert space question
Posted: Nov 10, 2012 11:46 PM
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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?
Pointers to good references are especially appreciated.
Many thanks in advance,
Dan



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