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

 Messages: [ Previous | Next ]
 Mike Terry Posts: 767 Registered: 12/6/04
Re: Dimension of the space of real sequences
Posted: Nov 13, 2012 7:00 PM

"José Carlos Santos" <jcsantos@fc.up.pt> wrote in message
news:agfvqlFbot6U1@mid.individual.net...
> Hi all,
>
> Can someone please tell me how to prove that the real vector space of
> all sequences of real numbers has uncountable dimension?
>
> Best regards,
>
> Jose Carlos Santos

You need to exhibit an uncountable set of vectors that are linearly
independent - i.e. no finite linear combination of the vectors can be zero.

I imagine there must be lots of ways to exhibit such a set, but as a hint
for the approach that occured to me: think "reals" (= "Dedekind cuts":
uncountably many of these...) composed from rationals (countably many of
these, like the countable number of terms in a sequence...).

Regards,
Mike.

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