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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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 Butch Malahide Posts: 894 Registered: 6/29/05
Re: Dimension of the space of real sequences
Posted: Nov 15, 2012 2:42 PM

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.

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