# Math Help - Showing vector space

1. ## Showing vector space

Hey,

I have given the space:

$l^{\infty}(\mathbb{N})=\Big\{\{x_{k}\}_{k=1}^{\inf ty} \vert x_{k}\in \mathbb{C}, \forall k\in \mathbb{N}, \text{and only finitely many} x_{k} \text{are nonzero} \Big\}$

How do I show that this space is a vector space. Should I run through various rules which must be fulfilled in order for a set to be a vector-space or is there another way of going about it?

Appreciate it.

2. Originally Posted by surjective
Hey,

I have given the space:

$l^{\infty}(\mathbb{N})=\Big\{\{x_{k}\}_{k=1}^{\inf ty} \vert x_{k}\in \mathbb{C}, \forall k\in \mathbb{N}, \text{and only finitely many} x_{k} \text{are nonzero} \Big\}$

How do I show that this space is a vector space. Should I run through various rules which must be fulfilled in order for a set to be a vector-space or is there another way of going about it?

Appreciate it.
It is confusing this notation. $l_\infty(\mathbb{N})$ is denoted usually by $l_\infty$ and it is the space of all bounded sequences. The space of sequences with only finite coordinates non-zero is the countable direct sum of copies of $\mathbb{C}$ and it is usually denoted by $\varphi$

To show that a set is a vector space simply show that the sum of two vectors defined in a natural way is in the set and also the multiplication by a scalar, it is simply algebra, and both, $l_\infty$ and $\varphi$ are vector spaces, the sum of two sequences which are bounded or eventually null is bounded or eventually null and the same with the multiplication by scalars.

3. Clearly the direct sum is a subset of direct product. So you'll only need to check that it's actually a subspace.

4. ## showi vector space

Hey,

Thanks for the reply. Yes I know that a vector space is a nonempty set V that has been equipped with two operations, namely addition and multiplication. The thing is that in order for a set to qualify as a vector space, the mentioned operations must adhere to certain axioms. They are quite many. Therefore I would, as far as possible, like to avoid showing all of them.

But as far as I have understood from your post it would be sufficient to choose two arbitrary bounded sequences in $l^{\infty}(\mathbb{N})$ and show that their sum is bounded as well as multiplying a scalar with an arbitrary bounded sequence in $l^{\infty}(\mathbb{N})$ and show that it is bounded. Would that really be sufficient to show that the space mentioned is a vector-space.

Appreciate the help.

5. Originally Posted by surjective
Hey,

Thanks for the reply. Yes I know that a vector space is a nonempty set V that has been equipped with two operations, namely addition and multiplication. The thing is that in order for a set to qualify as a vector space, the mentioned operations must adhere to certain axioms. They are quite many. Therefore I would, as far as possible, like to avoid showing all of them.

But as far as I have understood from your post it would be sufficient to choose two arbitrary bounded sequences in $l^{\infty}(\mathbb{N})$ and show that their sum is bounded as well as multiplying a scalar with an arbitrary bounded sequence in $l^{\infty}(\mathbb{N})$ and show that it is bounded. Would that really be sufficient to show that the space mentioned is a vector-space.

Appreciate the help.
Where's the confusion coming from? If I am understanding your question correctly the above space is a subset of a bigger vector space (which?) and to show that it's a vector space we must thus show it's closed under scalar mult. and addition.

But, if $x_n,y_n$ are eventually zero of order $k,\ell$ respectively then $x_n+y_n$ is eventually zero of order $\max\{k,\ell\}$. etc.

6. ## Showing vector space

Hey,

Let me try again. Looking at the previous posts I saw that I had written the space incorrectly. I have the space:

$l^{\infty}(\mathbb{N})=\Big\{\{x_{k}\}_{k=1}^{\inf ty} \vert x_{k}\in \mathbb{C}, \sup_{k \in \mathbb{N}}|x_{k}| < \infty \Big\}$

It does not state that $l^{\infty}(\mathbb{N})$´is a subset of a larger space!!!?

My confusion has to do with the textbook I'm using (and have been using since I started the course on Real Analysis). It is terrible and the formulation is not clear at all.

I know that a subspace of a vector-space $V$ is a subset $S\subseteq V$ such that when the addition and scalar multiplication of $V$ are used to add and scalar-multiply the elements of $S$, then $S$ is a vector-space. Hence by showing that $l^{\infty}(\mathbb{N})$ is closed under addition and multiplication then that would show the intended (as mentioned several times now).

Appreciate the help.

7. Originally Posted by surjective
Hey,

Let me try again. Looking at the previous posts I saw that I had written the space incorrectly. I have the space:

$l^{\infty}(\mathbb{N})=\Big\{\{x_{k}\}_{k=1}^{\inf ty} \vert x_{k}\in \mathbb{C}, \sup_{k \in \mathbb{N}}|x_{k}| < \infty \Big\}$

It does not state that $l^{\infty}(\mathbb{N})$´is a subset of a larger space!!!?

My confusion has to do with the textbook I'm using (and have been using since I started the course on Real Analysis). It is terrible and the formulation is not clear at all.

I know that a subspace of a vector-space $V$ is a subset $S\subseteq V$ such that when the addition and scalar multiplication of $V$ are used to add and scalar-multiply the elements of $S$, then $S$ is a vector-space. Hence by showing that $l^{\infty}(\mathbb{N})$ is closed under addition and multiplication then that would show the intended (as mentioned several times now).

Appreciate the help.
Where's the difficulty with this too? If $x_n\in\ell_\infty(\mathbb{N})$ then $\sup_{n\in\mathbb{N}}|\alpha x_n|=\alpha\sup_{n\in\mathbb{N}}|x_n|<\infty\impli es \{\alpha x_n\}\in\ell_{\infty}(\mathbb{N})$. Similarly, if $\{x_m\},\{y_n\}\in\ell_\infty(\mathbb{N})$ then $\sup_{n\in\mathbb{N}}|x_n+y_n|\leqslant\sup_{n\in\ mathbb{N}}|x_n|+\sup_{n\in\mathbb{N}}|y_n|<\infty\ implies \{x_n+y_n\}\in\ell_\infty(\mathbb{N})$

8. ## Showing vector space

Yes that is also what I was able to extract from the previous posts. Some general questions:

1) Is $l^{\infty}(\mathbb{N})$ a subset of a larger vector-space?

2) When you have a set equipped with the operations of addition and multiplication, is it then a given that these operations satisfy the wellknown axioms of a vector-space?

3) Does every supspace fulfill the properties of its larger vector-space?

$\text{thanks}^{\infty}$

9. Originally Posted by surjective
Yes that is also what I was able to extract from the previous posts. Some general questions:

1) Is $l^{\infty}(\mathbb{N})$ a subset of a larger vector-space?
Probably, you can probably embed it in a larger vector space. Is there a canonical choice? Probably not. Does it matter though? For if you are trying to embed it so you only have to prove closure of scalar mult. and addition (since it's a subspace) you'd still have to prove that the space you embedded it in is a vector space

2) When you have a set equipped with the operations of addition and multiplication, is it then a given that these operations satisfy the wellknown axioms of a vector-space?
What do you mean? Not every mult. and addition satisfies the vector space axioms.

3) Does every supspace fulfill the properties of its larger vector-space?
It inherits associativity and distributivity of the mult. and addition, etc. But it need not be closed under mult. and addition.

10. ## showing vector-space

I meant the following:

This is what my textbook says:

A vectorspace is a non-empty set V that has been equipped with two operations, called addition and scalar-multiplication, statisfying certain rules.

Notice the underlined. What you have helped me to show so far is that the space $l^{\infty}(\mathbb{N})$ is closed under addition and multiplication. But what about the "certain rules" mentioned? Is is not a requirement to show them also?

11. Originally Posted by surjective
I meant the following:

This is what my textbook says:

A vectorspace is a non-empty set V that has been equipped with two operations, called addition and scalar-multiplication, statisfying certain rules.

Notice the underlined. What you have helped me to show so far is that the space $l^{\infty}(\mathbb{N})$ is closed under addition and multiplication. But what about the "certain rules" mentioned? Is is not a requirement to show them also?
Yeah, but they are trivial. They are that...well I'm not going to write them all. Here they are. Personally, I would just say "it's apparent that they are satisfied" but I wouldn't do that if I were you.

12. I'm suspecting more problems will show up but thank a lot for now.