My questions is about Schauder bases and more specifically about coefficient functionals.

Let $(x_n)$ be a Schauder basis of a Banach space $X$. Thus for all $x$ in $X$, $x = \sum f_n(x) x_n$. The $f_n$ are called coefficient functionals. They are continuous. If $X$ is reflexive, they form a basis of $X^*$ (with Hahn–Banach theorem). My question is : if $X$ is not reflexive, but if we suppose $X^*$ separable, is $(f_n)$ a basis of $X^*$?