Let $X$ be reflexive space. Let $u_k\to u$ weakly. That means for every $\phi\in X^*$, we have $\phi(u_k)\to \phi(u)$. This, and the reflexivity imply that $u_k\to u$ point-wise in $X^{**}$. By uniform boundedness principle, we indeed have that $u_k$ is bounded uniformly in $X^{**}$. By isometry of $X$ and $X^{**}$, $u_k$ is bounded uniformly in $X$.
However it seems intuitively clear that weak convergence directly implies uniform boundedness. Maybe it holds only on convergence in norm topology. I might be wrong.
Could anyone help me clarify this?