```Date: Nov 12, 2012 2:53 PM
Author: Achimota
Subject: Re: countable set of closed subspaces in separable Hilbert space question

On Monday, November 12, 2012 3:52:24 PM UTC+8, William Elliot wrote:> 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 }"A+B" and "-A" are here ill-defined because + and - is not in general defined on sets. The definition would be valid if the definition specified a group (A,+) or the equivalent rather than simply that "A is a set". The case I am working with is linear subspaces, and in this case the definition > -A = { -a | a in A }is trivial because linear subspaces are closed under scalar-vector multiplication and so  a in A ==> -a in Awhich implies (under the above definition)  A = -Aand  A-B = A+BOn Monday, November 12, 2012 3:52:24 PM UTC+8, William Elliot wrote:> 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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