
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.

