# 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 $l_\infty(R)$ . If they are, verify by axioms. If not, provide counter example.

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

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

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

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

$y \in l_\infty(R)$

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

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

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

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

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

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

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

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 $l_\infty(R)$ . If they are, verify by axioms. If not, provide counter example.

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

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

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

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

$y \in l_\infty(R)$

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

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

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

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

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

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

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

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, $\|x\|_*=0\text{ if and only if }x=0~?$

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, $\|x\|_*=0\text{ if and only if }x=0~?$
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

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

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

Thanks

4. ## Re: Vector Space of Bounded Sequences

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

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

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

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

6. ## Re: Vector Space of Bounded Sequences

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