# Thread: can you help me?

1. ## subset R^w is a subspace?

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...

2. 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

3. 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?

4. 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.

5. 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

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

7. 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