I have the following question:
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 $F:X\longrightarrow\mathbb{R}$ a continuous function.
Is it true that $\quad\displaystyle\liminf_{n\to\infty}|f(x_n)|\;{\color{red}\geq}\;{\color{red}|}f(x){\color{red}|}$?
Recall that:
$\displaystyle\liminf_{n\to\infty}|f(x_n)|:=\sup_{n\in\mathbb{N}}\inf_{k\geq n}|f(x_k)|$.
Thanks in advance!