$\langle x,x_n\rangle \leq \|x_n\|$

Question

I am wondering if we have that $x_n\to x$ weakly, is it then true that $\langle x,x_n\rangle \leq \|x_n\|$, and why?

Answer 1

For me it is wrong.

On $\mathbb{R}$ take $x=2$ and $x_n = 2- \frac{1}{n}$.

Then $\langle x,x_n\rangle = 2\left(2- \frac{1}{n}\right)$, $\| x_n\|=2- \frac{1}{n} < 2\left(2- \frac{1}{n}\right)$

and of course $x_n \to x$ strongly so a fortiori weakly.