
Re: countable set of closed subspaces in separable Hilbert space question
Posted:
Nov 11, 2012 2:54 AM


> On Sat, 10 Nov 2012, Daniel J. Greenhoe wrote: > > Let H be a separable Hilbert space.
On Sunday, November 11, 2012 1:56:02 PM UTC+8, William Elliot wrote: > Vectorial subspaces or topological subspaces?
A Hilbert space is a special case of a linear space. The elements of the sets are called vectors, so in that sense one could call the subspaces "vector subspaces".
A Hilbert space is a complete inner product space. The inner product induces a norm which in turn induces a metric which in turn induces a topology on the space. So, in that sense one could also call the subspaces "topological subspaces".
The conclusion is this: In a Hilbert space, there is no distinction between a "vector subspace" and a "topological subspace". All subspaces in a Hilbert space are both "vectorial" (in the sense they contain vectors) and "topological" (in the sense that the inner product induces a topology).
Dan
On Sunday, November 11, 2012 1:56:02 PM UTC+8, William Elliot wrote: > 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?

