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- Oct 23rd 2009, 03:29 PM #1

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## subset R^w is a subspace?

I have trouble to prove that

Is the following subset (the vector space of real-valued sequences, with vector addition and scalar multiplication defined as for ) a subspace?

D={a | the entries a are limited to finitely many values}

I appreciate to help...thank you...

- Oct 23rd 2009, 03:52 PM #2

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- Oct 23rd 2009, 06:08 PM #3

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I would interpret "the entries are limited to finitely many values" to mean "there are only a finite number of real numbers that you can have as entries" and I think that is how Tonio interpreted it.

But I suspect you mean "only a finite number of the entries are non-zero" which is a very different thing. The former is NOT a subspace (for the reason Tonio suggested) and the latter is.

Which do you mean?

- Oct 25th 2009, 01:11 AM #4

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A sequence a will belong to GAMMA iff there is a set of values M=

{r1...rk}, such that ai must come from M

Thus <1,2,2,1,2,2,1,...> would belong to GAMMA since all of its

entries come from the finite set {1,2}. NOTE: This does not mean that

members of GAMMA need to have a repeating pattern. It simply means

that the set of numbers which occur in the sequence is finite.

- Oct 25th 2009, 05:11 AM #5

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- Oct 25th 2009, 08:43 AM #6

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- Oct 25th 2009, 09:54 AM #7

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