W^3
Posts:
29
Registered:
4/19/11


Re: Dimension of the space of real sequences
Posted:
Nov 15, 2012 5:14 PM


In article <903909e246734c2cbe09e1be2da87102@y8g2000yqy.googlegroups.com>, Butch Malahide <fred.galvin@gmail.com> wrote:
> 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.
Let's do this instead: l^2 is isomorphic to L^2([0,2pi]) (as vector spaces and much more), and the set {Chi_(0,t) : t in (0,2pi)} is linearly independent in L^2([0,2pi]).

