Date: Nov 13, 2012 7:00 PM
Author: Mike Terry
Subject: Re: Dimension of the space of real sequences

"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.