can you help me?

**iloveyou7** · October 23rd 2009, 03:29 PM

I have trouble to prove that

Is the following subset $R^w$ (the vector space of real-valued sequences, with vector addition and scalar multiplication defined as for $R^n$ ) a subspace?
D={a

$R^w$ | the entries a $i$ are limited to finitely many values}

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

**tonio** · October 23rd 2009, 03:52 PM

Originally Posted by iloveyou7

I have trouble to prove that

Is the following subset $R^w$ (the vector space of real-valued sequences, with vector addition and scalar multiplication defined as for $R^n$ ) a subspace?
D={a

$R^w$ | the entries a $i$ are limited to finitely many values}

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

How many possible scalars can multiply an element in D?

Tonio

**HallsofIvy** · October 23rd 2009, 06:08 PM

Originally Posted by iloveyou7

I have trouble to prove that

Is the following subset $R^w$ (the vector space of real-valued sequences, with vector addition and scalar multiplication defined as for $R^n$ ) a subspace?
D={a

$R^w$ | the entries a $i$ are limited to finitely many values}

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

I would interpret "the entries $a_i$ 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 $a_i$ 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?

**iloveyou7** · October 25th 2009, 01:11 AM

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.

**tonio** · October 25th 2009, 05:11 AM

Originally Posted by iloveyou7

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.

Ok, and this is what I understood from your question and thus my hint answers it.

Tonio

**iloveyou7** · October 25th 2009, 08:43 AM

But i don't know how to prove it...

**tonio** · October 25th 2009, 09:54 AM

Originally Posted by iloveyou7

But i don't know how to prove it...

Take your example with all the sequences whose entries can be only 1 or 2. If this is a subspace, then it is closed under multiplication by scalar. Take, for example, the sequence $\left\{1,2,1,2,1,2....\right\}$ and multiply it by the scalar -1: what happens?

Tonio

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