how $||T||=1$ in $(\eta_j)=T(x), \eta_j=\frac{\zeta_j}{j}, x=(\zeta_j)$?

Question

In Kreyszig's Functional Analysis it says that for $T: l^{\infty} \to l^{\infty}$ defined by $y=(\eta_j)=T(x), \eta_j=\frac{\zeta_j}{j}, x=(\zeta_j)$, we have $||T|| := \sup_{||x|| (\ne 0) \in l^{\infty}} \dfrac{||Tx||}{||x||}=1$. I can't figure that out how $||T||=1$? Please help! Thanks.

Answer 1

Let $e_1=(1,0,0,...)$ Then $Te_1=e_1$ and $\|e_1 \|=1$. Hence $\sup_{x \neq 0} \frac {\|Tx\|} {\|x\|} \geq \frac {\|Te_1\|} {\|e_1\|}=1$. On the other hand $\frac 1 j \leq 1$ for all $j$ so $\|Tx\|\leq \|x\|$ for all $x$. Hence $\sup_{x \neq 0} \frac {\|Tx\|} {\|x\|}$ is exactly $1$.