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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
Re: countable set of closed subspaces in separable Hilbert space question
Posted: Nov 12, 2012 2:20 AM
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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?





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