From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: normed linear vector spaces
Date: 17 Jul 2001 07:40:58 -0400
Newsgroups: sci.math
Summary: Possible topologies for a finite-dimensional topological vector space

Michele Dondi <bik.mido@tiscalinet.it>: 

>>> I know that "on finite dimensional vector spaces all NORMS are
>>> equivalent" i.e. they generate the same topology. On the other hand
>>> there are topologies compatible with the vector space structure that
>>> are different from euclidean topology. For example the discrete
>>> topology, that (is a metric topology, but) is not a normizable
>>> topology.

Stephen Montgomery-Smith <stephen@math.missouri.edu>:
>>Well when I said topology, I was speaking loosely, and I meant topology
>>compatible with the vector space.  
...
>>That would rule out the
>>discrete topology.
>
>D'Oh! Yet there must be some _simple_ example that DOES work. I mean a
>topology that makes a finite-dimensional linear space V into a
>topological linear space and that is not the euclidean topology (up to
>a homeomorphism). Of course, by Tychonov's theorem, such a topology
>can't be Hausdorff.

In any topological vectorspace E (=vectorspace over a topological field
equipped with what Stephen has called a "topology compatible with the
vector space" structure), the closure of the point 0 is a closed 
subvectorspace K, and the quotient vectorspace E/K with its quotient
topology is again a topological vectorspace, in which the subspace
{0} (and therefore any one-point set) is closed.  But a topological
vectorspace in which points are closed is Hausdorff; so, if E (and
therefore E/K) is a finite-dimensional real or complex vectorspace, 
E/K is linearly homeomorphic to a real or complex Euclidian space.
Conversely, if E is any real or complex vectorspace, and K is any
subspace such that E/K is finite-dimensional, then E can be given
the weakest topology (fewest open sets) such that K is closed and
the quotient topology on E/K is linearly homeomorphic to the 
Euclidian topology.  Restricted to finite-dimensional E, this
classifies topological vectorspaces.  In particular, every
finite-dimensional topological vectorspace is linearly homeomorphic 
to the product of a Euclidian vectorspace and a vectorspace with
the indiscrete topology (only two open sets).

The simplest "simple example" sought above is therefore R with the
indiscrete topology; the second-simplest is the product of the usual
R with the indiscrete R.

Lee Rudolph