Let $X$ be a Banach space and let $x_n \overbrace{\rightarrow}^w x$ and $x_n \overbrace{\rightarrow}^s z$ can we then say that $x = z$? My try:
$\| x- z\| = \sup_{\ell \leq 1} |\ell(x-z)| = \sup_{\ell \leq 1} |\ell x -\ell z| \leq \epsilon$
Where $\ell$ is a continuous functional in $X'$ Is this correct? is there any easier way? Thanks Btw if this already is correct, should I delete the post or what do I do?