So here is my question,
In a exercise I have to decide if the following holds,
Let $X$ be a normed vectorspace and let $x_n\rightharpoonup x$ i.e $(x_n)_{n\in\mathbb N}$ converges weak to $x$. And let $f_n\rightarrow f$ for $f_n,f\in X^*$.
Then, $$f_n(x_n)\rightarrow f(x)$$
I am not sure if this is true but I think the statement is wrong since everytime I assumed it is correct and started to prove it, I arrived somewhere, where strong convergence of $(x_n)_{n\in\mathbb N}$ was required to show that the statement holds... So I even think that this implies strong convergence of $(x_n)_{n\in\mathbb N}$.
Could someone help me and tell me if I am correct? Thanks!
Note that a weakly convergent sequence is bounded, so there is a $C < \infty$ with $\lVert x_n\rVert \leqslant C$ for all $n$. Further,
$$f_n(x_n) - f(x) = \bigl(f_n(x_n) - f(x_n)\bigr) + \bigl( f(x_n) - f(x)\bigr).$$
The weak convergence $x_n \rightharpoonup x$ and the norm convergence $f_n \to f$ together are sufficient to conclude $f_n(x_n) \to f(x)$.