W^3
Posts:
28
Registered:
4/19/11
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Re: Dimension of the space of real sequences
Posted:
Nov 15, 2012 5:14 PM
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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]).
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