Date: Nov 13, 2012 7:00 PM
Author: Mike Terry
Subject: Re: Dimension of the space of real sequences
"José Carlos Santos" <firstname.lastname@example.org> wrote in message
> 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...).