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Topic: Dimension of the space of real sequences
Replies: 21   Last Post: Nov 19, 2012 10:22 AM

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 Jose Carlos Santos Posts: 4,896 Registered: 12/4/04
Re: Dimension of the space of real sequences
Posted: Nov 14, 2012 2:22 AM

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

Date Subject Author
11/13/12 Jose Carlos Santos
11/13/12 Mike Terry
11/14/12 Jose Carlos Santos
11/14/12 Mike Terry
11/13/12 Ken.Pledger@vuw.ac.nz
11/13/12 Virgil
11/14/12 Jose Carlos Santos
11/14/12 Shmuel (Seymour J.) Metz
11/13/12 archimede plutanium
11/14/12 Robin Chapman
11/14/12 David Bernier
11/14/12 Jose Carlos Santos
11/14/12 Robin Chapman
11/14/12 Jose Carlos Santos
11/14/12 quasi
11/14/12 Jose Carlos Santos
11/14/12 W^3
11/15/12 David C. Ullrich
11/15/12 Butch Malahide
11/15/12 W^3
11/18/12 David Bernier
11/19/12 David C. Ullrich