Let $X$ be a Banach space and let $M: X \rightarrow X$ be a linear map. Prov that M is bounded iff there exists a set $S \subset X'$, dense in X', such that for each $\ell \in S$ the functional $m_l$ defined by $$m_\ell(x) = \ell M (x)$$ is continuous on X.
My try: If $M$ is bounded then $\ell M$ is bounded for all $\ell$, hence all $m_\ell$ are continuous, for all subsets $S$. So we need to find a dense one?
On the other hand: suppose all $m_\ell$ is continuous, since the weak limit is unique, $Mx_n \rightarrow y$ and $x_n \rightarrow x$ $\Longrightarrow$ $Mx = y$ and by the closed graph M is bounded/continuous.
It feels like I'm missing something with the denseness of $S$. Should I look at $\ell \in S^c$ also? and do some $\epsilon/2$ argument?