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...?