# Weak limit and strong limit

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?

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What is $\varepsilon$? And in the supremum, you should write you take it over the linear continuous functionals of norm $\leqslant 1$. – Davide Giraudo Dec 29 '12 at 14:43
$\epsilon$ is arbitrary small number? Will the choice of $\ell$ affect my result? – Johan Dec 29 '12 at 14:55
No problem as the norm is bounded by $1$. But the las inequality deserves an additional step introducing the sequence $\{x_n\}$ (hence you don't really need $\varepsilon$). – Davide Giraudo Dec 29 '12 at 14:59

Yes. If $x_{n} \to z$ strongly, then $x_{n} \to z$ weakly also. Since the weak topology is Hausdorff, we have uniqueness of the limit. Therefore, $z = x$.
If $x_n\to x$ weakly and $x_n\to z$ weakly, it follows that $f(x)=f(z)$ for any $f\in X^{*}$. Hence $f(x-z)=0$ for any $f\in X^{*}$. An immediate application of Hahn-Banach implies $x=z$. Indeed, otherwise we can build a functional $g_0$ on span$\{x-z\}$ such that $g_0(x-z)=1$, and use Hahn-Banach extend it to a functional $g$ on $X$. For this $g\in X^{*}$ we would have $g(x-z)=1$.