From: Yossi Lonke <jrl16@po.cwru.edu>
Subject: Re: Topology??
Date: Wed, 24 Jan 2001 17:01:27 -0500
Newsgroups: sci.math.research
Summary: Topology on limits of topological spaces

One way is this: a sequence s_k with each s_k in X_k will tend to zero,
if sup_k{N_k(s_k)} = 0, where N_k is the semi norm on X_k.
Obviosuly if the limit of any sequence of that form exists, it
must be in X. So in general we'll say that s_k tends to s,
if sup_k{N_k(s_k-s)} = 0, and this is well defined since s is in X_k for
all k.

Hope this helps.

Yossi Lonke

> From: "Rob Brownlee" <rab14@leicester.ac.uk>
> Organization: University of Illinois at Urbana-Champaign
> Newsgroups: sci.math.research
> Date: Wed, 24 Jan 2001 15:43:23 -0000
> Subject: Topology??
> 
> I have the following scale of spaces:
> 
> L^2=X_0 \supset X_1 \supset X_2 \supset ... \supset X_k \supset X_{k+1}
> \supset ... \supset X \subset Y.
> 
> Where X is the intersection of all X_k.
> 
> I have a sequence {s_k}with each x_k \in X_k. I'd like to talk about s_k ->
> s, where s is my candidate "limit". In what sense can I talk about
> convergence? I'm entertaining any suggestion that anyone can think of. What
> maybe relevent is that each X_k carries a semi-norm and the topology on X
> induced by the family of these semi-norms restircted to Y conincides with a
> topology on Y induced by a norm. Any help welcomed.
> 
> Rob