Let $X$ be a Banach space and $X^*$ be its dual space. Let $\phi_n\in X^\ast$ and for all $x\in X$ we have $\phi_n(x)\to c\in\mathbb{C}$ as $n\to\infty$. I want to show that the sequence $\phi_n$ has a weak$^*$ limit $\phi\in X^*$.
Also if $x_n$ is a sequence in $X$ and for all $\phi\in X^*$ we have $\phi(x_n)\to a\in\mathbb{C}$. I want to show $x_n$ converges weakly in $X$ if $X$ is reflexive.
Thank you!