Consider a Banach $\mathbb{R}$-space E and an element $u\in E''$ such that : for all sequence $\phi_n\in E'$ which $\sigma(E',E)$-converges to $\phi \in E'$, one has $\lim u(\phi_n)=u(\phi)$. Is it true that there is some $x\in E$ such that $u(f)=f(x)$ for all $f\in E'$ ?

remark : it is not clear for me that $u:E'\rightarrow \mathbb{R}$ is $\sigma(E',E)$-continuous