Prove a the convergence of a series in dual Banach Space

Question

Suppose we have a series where $\{u_{n}\}_{n=1}^{\infty}\subset X'$, $u\in X'$, and this satisfies $u_{n}\xrightarrow{w^{*}} u.$ Where X is the Banach Space over $F$. Is $\|u\|_{X'}\leq \lim\limits inf $ ${n \to \infty} \|u_{n}\|_{X'}?$

Answer 1

By uniform boundedness principle it follows that $(u_n)$ is bounded in $X'$. Moreover, $$ u(x) = \lim_{n\to \infty} u_n(x) $$ implies $$ |u(x)| \le \lim\inf_{n\to\infty} |u_n(x)| \le \lim\inf_{n\to\infty} \|u_n\|_{X'} \|x\|, $$ which shows $$ \|u\|\le \lim\inf_{n\to\infty} \|u_n\|_{X'} . $$