I have the following question (a variation from Property of limit inferior for continuous functions):
Let $(x_n)$ a sequence in $X$ and $x\in X$ such that for all $F\in X'$ (the dual space of the vector space X) we have that $(F(x_n))$ converge to $F(x)$ (that is: the sequence converge weakly on $X$).
Let $F\color{red}{\in X'}$.
Is it true that $\quad\color{red}\|\color{red}F\color{red}\|\displaystyle\liminf_{n\to\infty}{\,\color{red}\| \color{red}x_{\color{red}n}\color{red}\|}\;{\color{red}\geq}\;{\color{red}|}F(x){\color{red}|}$?
Recall that:
$\displaystyle\liminf_{n\to\infty}\|x_n\|:=\sup_{n\in\mathbb{N}}\inf_{k\geq n}\|x_k\|$.
Thanks in advance!