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October 1st, 2009, 02:51 PM   #1
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Prove S is not finitely generated. please help

Let S={(a1,a2,a3,...)l ai c-(belongs to) R, forall i c- N} be the vector space of a infinite sequences of real numbers. (We know that S is a vector space over the feild R.) Prove carefully that S is not finitely generated.

Can a spanning set equal all of the elements of S? Then the largest spanning set would be infinite because S has no limit to the elements in it. If I am wrong can someone please prove this or if I'm right and you would like to prove this as well, please be my guess. thank you
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October 1st, 2009, 03:59 PM   #2
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Re: Prove S is not finitely generated. please help

Quote:
Originally Posted by mattpi
Suppose is a finite set that is contained in S. Since S contains only finite sequences (I am taking this to mean that for each there exists an such that for all ), each has a highest non-zero term. Let t_i)_n\neq0\text{ for some }i)." /> Then the sequence where the '1' is in the (K+1)th position cannot be expressed as a linear combination of the elements of Therefore T does not span S.
OK - given the original typo, that's all irrelevant now...

Suppose there exists a finite basis Consider the vectors etc.

The set is contained in and is linearly independent (easy to show). It is of dimension larger than the basis Can you see why cannot be a basis?
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