The problem is:
Let $E$ be a normed space, let $x\in E$, let $\left(x_{n}\right)_{n\in\mathbb{N}}\subset E$ be a sequence. I need to show that $\left(x_{n}\right)$ converges weakly to $x$ if and only if $\left(x_{n}\right)$ converges to $x$ in $\left(E,\,\sigma\left(E,\, E^{*}\right)\right)$, $\sigma\left(E,\, E^{*}\right)$ denoting the weak topology on $E$.
My efforts:
I have that $x_{n}\rightarrow x$ in $\left(E,\,\sigma\left(E,\, E^{*}\right)\right)$ is equivalent to: $\forall U\in\sigma\left(E,\, E^{*}\right),\, x\in U\,:\quad\left\{ n\in\mathbb{N}:\, x_{n}\notin U\right\}$ is finite.
My question:
It is unclear to me how to pass from the weak convergence in $E$ to the convergence in the weak topology on $E$.
How can I prove both directions?
Thanks, Franck.