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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!

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  • $\begingroup$ I would say yes, since $\liminf_n \|x_n \| \geq \|x\|$ and $\|F\|_{op} \|x\| \geq |F(x)|$. $\endgroup$ – Xiao Sep 24 '14 at 21:13
  • $\begingroup$ I meant the dual norm of $F$ instead of the operator norm. $\endgroup$ – Xiao Sep 24 '14 at 22:00

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