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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
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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
Read Dimension of the space of real sequences
Jose Carlos Santos
11/13/12
Read Re: Dimension of the space of real sequences
Mike Terry
11/14/12
Read Re: Dimension of the space of real sequences
Jose Carlos Santos
11/14/12
Read Re: Dimension of the space of real sequences
Mike Terry
11/13/12
Read Re: Dimension of the space of real sequences
Ken.Pledger@vuw.ac.nz
11/13/12
Read Re: Dimension of the space of real sequences
Virgil
11/14/12
Read Re: Dimension of the space of real sequences
Jose Carlos Santos
11/14/12
Read Re: Dimension of the space of real sequences
Shmuel (Seymour J.) Metz
11/13/12
Read Re: Dimension of the space of real sequences
archimede plutanium
11/14/12
Read Re: Dimension of the space of real sequences
Robin Chapman
11/14/12
Read Re: Dimension of the space of real sequences
David Bernier
11/14/12
Read Re: Dimension of the space of real sequences
Jose Carlos Santos
11/14/12
Read Re: Dimension of the space of real sequences
Robin Chapman
11/14/12
Read Re: Dimension of the space of real sequences
Jose Carlos Santos
11/14/12
Read Re: Dimension of the space of real sequences
quasi
11/14/12
Read Re: Dimension of the space of real sequences
Jose Carlos Santos
11/14/12
Read Re: Dimension of the space of real sequences
W^3
11/15/12
Read Re: Dimension of the space of real sequences
David C. Ullrich
11/15/12
Read Re: Dimension of the space of real sequences
Butch Malahide
11/15/12
Read Re: Dimension of the space of real sequences
W^3
11/18/12
Read Re: Dimension of the space of real sequences
David Bernier
11/19/12
Read Re: Dimension of the space of real sequences
David C. Ullrich

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