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