# Thread: Vector Space of Bounded Sequences

1. ## Vector Space of Bounded Sequences

Folks,

Consider the vector space l_infty(R) of all bounded sequences. Decide whether or not the following norm is defined on $\displaystyle l_\infty(R)$ . If they are, verify by axioms. If not, provide counter example.

$\displaystyle x \in l_{\infty} (R); x=(x_n)$,

(i) $\displaystyle || ||_*$ defined by $\displaystyle ||x||_*=|x_1|$

a) Since $\displaystyle x \in l_\infty(R)$

$\displaystyle \exists k_x >0$ s.t $\displaystyle |x_n| \le k_x \forall n \in N$

$\displaystyle y \in l_\infty(R)$

$\displaystyle \exists k_y>0$ s.t $\displaystyle |y_n| \le k_y \forall n \in N$

$\displaystyle ||x+y||_*=||(x_n+y_n)||_*=|x_n+y_n|$

Now we have that $\displaystyle |a+b| \le |a|+|b|$, therefore

$\displaystyle |x_n+y_n|\le|x_n|+|y_n|$

$\displaystyle \le k_x+k_y \forall n \in N$

$\displaystyle \le ||x||_*+||y||_*$

b) $\displaystyle ||ax||_*=|ax_1|=|a| |x_1|=|a |||x||_*$

Axioms c) and d) I dont know how to attempt for this space...?

2. ## Re: Vector Space of Bounded Sequences

Originally Posted by bugatti79

Consider the vector space l_infty(R) of all bounded sequences. Decide whether or not the following norm is defined on $\displaystyle l_\infty(R)$ . If they are, verify by axioms. If not, provide counter example.

$\displaystyle x \in l_{\infty} (R); x=(x_n)$,

(i) $\displaystyle || ||_*$ defined by $\displaystyle ||x||_*=|x_1|$

a) Since $\displaystyle x \in l_\infty(R)$

$\displaystyle \exists k_x >0$ s.t $\displaystyle |x_n| \le k_x \forall n \in N$

$\displaystyle y \in l_\infty(R)$

$\displaystyle \exists k_y>0$ s.t $\displaystyle |y_n| \le k_y \forall n \in N$

$\displaystyle ||x+y||_*=||(x_n+y_n)||_*=|x_n+y_n|$

Now we have that $\displaystyle |a+b| \le |a|+|b|$, therefore

$\displaystyle |x_n+y_n|\le|x_n|+|y_n|$

$\displaystyle \le k_x+k_y \forall n \in N$

$\displaystyle \le ||x||_*+||y||_*$

b) $\displaystyle ||ax||_*=|ax_1|=|a| |x_1|=|a |||x||_*$

Axioms c) and d) I dont know how to attempt for this space...?
I am not sure what you mean by "Axioms c) and d)"

But have you considered, $\displaystyle \|x\|_*=0\text{ if and only if }x=0~?$

3. ## Re: Vector Space of Bounded Sequences

Originally Posted by Plato
I am not sure what you mean by "Axioms c) and d)"

But have you considered, $\displaystyle \|x\|_*=0\text{ if and only if }x=0~?$
I shouldnt have labelled them c and d. I just meant the 4 general axioms of a norm.
I think what you have wrote is one of them.

is the final one

$\displaystyle ||x||_*=|x_1| \ge 0$ iff $\displaystyle |x_1| \ge 0$

$\displaystyle \forall (x_n) \in R$..?

Thanks

4. ## Re: Vector Space of Bounded Sequences

Originally Posted by bugatti79
I just meant the 4 general axioms of a norm.
$\displaystyle ||x||_*=|x_1| \ge 0$ iff $\displaystyle |x_1| \ge 0$
$\displaystyle \forall (x_n) \in R$..?
That is what confused me. Usually we just use three properties for norm.

Let $\displaystyle t=(0,1,1,\cdots,1\cdots)$ what is $\displaystyle \|t\|_*=~?$

5. ## Re: Vector Space of Bounded Sequences

Originally Posted by Plato
That is what confused me. Usually we just use three properties for norm.

Let $\displaystyle t=(0,1,1,\cdots,1\cdots)$ what is $\displaystyle \|t\|_*=~?$
Well since $\displaystyle ||||_*$ is defined by $\displaystyle ||x||_*=|x_1|$

Then $\displaystyle \|t\|_*=|0|=0$...?

6. ## Re: Vector Space of Bounded Sequences

Originally Posted by bugatti79
Well since $\displaystyle ||||_*$ is defined by $\displaystyle ||x||_*=|x_1|$
Then $\displaystyle \|t\|_*=|0|=0$...?
Well can $\displaystyle \bold{t}=\bold{0}~?$