Sorry to bother but I'm having some problems proving some properties in my way to prove some others weak and weak-* metrizability properties.
First If I got a Banach space $X$, wich its dual $X^{*}$ is separable. Then the closed unit ball of the dual is separable with respect to the norm of $X^{*}$ which is the usual.
In this case, I think I have to find a dense and countable family which fulfills what we need
Second Once we prove the above statement, lets suppose that the countable dense family we found is $\phi$ contained in the closed unit ball of the dual. Then if we define:
$d(x,y)=\sum_{k=1}^{\infty} 2^{-k}|\phi_{k}(x-y)|$ for $x,y\in X$
Im having problems proving that is in fact a metric. In particular, I'm struggling with the implication $d(x,y)=0 \implies x=y$.
Thanks so much for your help!
Let $(\phi_k)$ be dense in the unit ball of the dual w.r.t. weak* topology. Then $d(x,y)=0$ implies $\phi_k( x)=\phi_k(y)$ for all $k$. This implies $\phi (x)=\phi(y)$ for every $\phi $ in the unit ball of the dual hence for every $\phi$ in the dual space. This implies $x=y$ because $x-y \neq 0$ implies that there exists $\phi$ with $\phi(x-y) \neq 0$. [This is a consequence of Hahn Banach Theorem].