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Re: Dimension of the space of real sequences
Posted:
Nov 14, 2012 2:22 AM
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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
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