Date: Nov 14, 2012 2:22 AM
Author: Jose Carlos Santos
Subject: Re: Dimension of the space of real sequences
On 14-11-2012 0:00, Mike Terry wrote:

>> Can someone please tell me how to prove that the real vector space of

>> all sequences of real numbers has uncountable dimension?

>

> 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...).

Great hint. Thanks.

Best regards,

Jose Carlos Santos