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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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 David C. Ullrich Posts: 21,553 Registered: 12/6/04
Re: Dimension of the space of real sequences
Posted: Nov 15, 2012 8:44 AM

On Wed, 14 Nov 2012 18:19:29 -0800, W^3 <82ndAve@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.

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