let $\{\phi_k\}$ be a complete orthonormal system in $L^2$. let $m=\{m_k\}$ be a sequence and let $f\in L^2$, $f\sim \sum c_k\phi_k$.
define an oparator $T$ by $Tf\sim \sum m_k c_k \phi_k$. then, show that
$m\in l^\infty\iff \exists c:const$ s.t. $||Tf||_{L^2} \leq c||f||_{L^2}$ $\forall f\in L^2$,
in particular, the smallest $c$ is $||m||_{l^\infty}$.
I could only show $\implies$ by letting $c=||m||_{l^\infty}$, but have no clue about the other parts.
any hints will be appriciated!
Note that $$\tag{1} |m_k|=\|T\phi_k\|_2. $$ Then $$ |m_k|=\|T\phi_k\|_2\leq c\|\phi_k\|_2=c. $$ So $m\in\ell^\infty$ and $\|m\|\leq c$.
It might help to understand what needs to be done if you work in $\ell^2$ instead of $L^2$ (those two are isomorphic via $$ (x_k)\longmapsto \sum_kx_k\,\phi_k. $$