Date: Nov 11, 2012 2:54 AM
Author: Achimota
Subject: Re: countable set of closed subspaces in separable Hilbert space question
> 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?