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


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 illdefined. In fact, I don't even have a definition for XX_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 > > >  xx_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?

