
Re: Dimension of the space of real sequences
Posted:
Nov 15, 2012 8:44 AM


On Wed, 14 Nov 2012 18:19:29 0800, W^3 <82ndAve@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.

