Date: Nov 15, 2012 8:44 AM
Author: David C. Ullrich
Subject: Re: Dimension of the space of real sequences

On Wed, 14 Nov 2012 18:19:29 -0800, W^3 <> 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.