Date: Nov 15, 2012 2:42 PM
Author: Butch Malahide
Subject: Re: Dimension of the space of real sequences

On Nov 15, 7:44 am, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> On Wed, 14 Nov 2012 18:19:29 -0800, W^3 <82nd...@comcast.net> wrote:
> >If R^N had a countable basis, then so would every subspace of R^N. In
> >particular l^2 would have a countable basis, call it {v_1,_2, ...}.
> >Setting V_n = span {v_1, ..., v_n}, we then have l^2 = V_1 U V_2 U ...
> >But this violates Baire, as l^2 is complete (in its usual metric) and
> >each V_n is closed and nowhere dense in l^2.

>
> Very good. I thought there should be something more analytic or
> cardinalitic instead of the (very nice) algebraic trickery that's
> been given.


However, it seems to me that the "algebraic trickery" shows that there
is no basis of cardinality less than the continuum, whereas using
Baire category only shows that there is no countable base.