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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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