To find norm of a bounded linear operator

Question

I have given $S:l^{\infty} \to l^1$ by $S(x_1, x_2, x_3,...)=(\frac{x_1}{2},\frac{x_1+x_2}{2^2},\frac{x_1+x_2+x_3}{2^3},....)$. I showed it is bounded linear operator but I am not getting idea about how to find $\mid\mid S \mid\mid$. Any start or hint. Thanks

Answer 1

If $s\in\ell^\infty$ and $\|s\|\leqslant1$, then $s=(x_1,x_2,x_2,\ldots)$ and $(\forall n\in\Bbb N):|x_n|\leqslant1$. So\begin{align}\bigl\|S(s)\bigr\|_1&=\sum_{n=1}^\infty\frac{|x_1+x_2+\cdots+x_n|}{2^n}\\&\leqslant\sum_{n=1}^\infty\frac{|x_1|+|x_2|+\cdots+|x_n|}{2^n}\\&\leqslant\sum_{n=1}^\infty\frac n{2^n}\\&=2.\end{align}So, $\|S\|\leqslant2$. But $\|(1,1,1,\ldots)\|_\infty=1$ and $\bigl\|S\bigl((1,1,1,\ldots)\bigr)\bigr\|_1=2$. So, $\|S\|=2$.