Date: Nov 15, 2012 5:14 PM
Author: W^3
Subject: Re: Dimension of the space of real sequences
In article

<903909e2-4673-4c2c-be09-e1be2da87102@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]).