Weak-*-Topology and sequences

Question

Let $X$ be a Banach Space , $x\in X$ and $(x^{*}_{n})_{n\in\mathbb{N}}$ a sequence in $X'$ with a weak-$*$-clusterpoint $x^{*}$. Does this imply that $x^{*}(x)$ is a clusterpoint of $(x^{*}_{n}(x))_{n\in\mathbb{N}}$ (which is a sequence in $\mathbb{R}$)? I have difficulties with this topology beacause i don't know which norm or metric i have to use (if there is any). How does a neighbourhood in the weak-*-topology look like?

Answer 1

Weak* topology is neither normable nor metrizable. A basic neighborhood of $x^{*}$ is of the form $\{y^{*}: |y^{*}(x_i)-x^{*}(x_i)|<r_i \forall i \in \{1,2,...,N\}\}$ for some integer $N$, some positive numbers $r_i$ and some vectors $x_1,x_2,..,x_n \in X$. To answer your question take a neighborhod with $N=1$ and $x_1=x$.