
Re: countable set of closed subspaces in separable Hilbert space question
Posted:
Nov 12, 2012 2:52 AM


On Sun, 11 Nov 2012, Daniel J. Greenhoe wrote: > 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.
In general, if A is a set and f a function f(A) is defined as the set { f(A)  a in A }. Accordingly,
A + B = { a + b  a in A, b in B } A = { a  a in A }
and perhaps, if ever used, A = { f : f in A }
would be a collection of numbers and as such one can't write A < r without the special definition A < r when for all a in A, a < r.
By special, I mean I use A <= r when for all a in A, a <= r; a definition not used by others. I don't use A < r.

