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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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William Elliot

Posts: 1,664
Registered: 1/8/12
Re: countable set of closed subspaces in separable Hilbert space
question

Posted: Nov 12, 2012 2:52 AM
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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 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.


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.




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