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Topic: Dimension of the space of real sequences
Replies: 21   Last Post: Nov 19, 2012 10:22 AM

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 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
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]).

Date Subject Author
11/13/12 Jose Carlos Santos
11/13/12 Mike Terry
11/14/12 Jose Carlos Santos
11/14/12 Mike Terry
11/13/12 Ken.Pledger@vuw.ac.nz
11/13/12 Virgil
11/14/12 Jose Carlos Santos
11/14/12 Shmuel (Seymour J.) Metz
11/13/12 archimede plutanium
11/14/12 Robin Chapman
11/14/12 David Bernier
11/14/12 Jose Carlos Santos
11/14/12 Robin Chapman
11/14/12 Jose Carlos Santos
11/14/12 quasi
11/14/12 Jose Carlos Santos
11/14/12 W^3
11/15/12 David C. Ullrich
11/15/12 Butch Malahide
11/15/12 W^3
11/18/12 David Bernier
11/19/12 David C. Ullrich