Date: Nov 12, 2012 2:20 AM
Author: Achimota
Subject: Re: countable set of closed subspaces in separable Hilbert space question
On Monday, November 12, 2012 11:21:30 AM UTC+8, William Elliot wrote:

> How for example, are you defining

> ||X - X_n|| for sets X and X_n?

William, I think you bring up a very good point. My "definition" of the limit of a sequence of subspaces is ill-defined. In fact, I don't even have a definition for X-X_n, and am not sure a good way to define the norm ||Y|| of a subspace Y.

Maybe that is why some authors use the uncountable union of subset condition in their definition of MRAs in wavelet analysis.

Thank you very much, I appreciate your help

Dan

On Monday, November 12, 2012 11:21:30 AM UTC+8, William Elliot wrote:

> On Sun, 11 Nov 2012, Daniel J. Greenhoe wrote:

>

>

>

> > On Sunday, November 11, 2012 5:09:34 PM UTC+8, William Elliot wrote:

>

> > > ...Do you mean the topological closure or some algebra construction?

>

> >

>

> > > How are you defining lim(n->oo) X_n?

>

>

>

> > "Strong convergence" ("convergence in the norm"); that is, the norm

>

> > induced by the inner product:

>

>

>

> > For any e>0 there exists N such that

>

> > || x-x_n || < e for all n>N

>

> >

>

> That's the usual definition for a sequence of points

>

> to converge to a point. It has nothing to do with

>

> a sequence of subsets or subspaces converging to a

>

> set or subspace. How for example, are you defining

>

> ||X - X_n|| for sets X and X_n?