topological vector spaces – Weak$^*$ continous functionals

Let $X$ be a Banach space (not necessarily reflexive) and let $X^{*}$ denote the continuous dual of $X$. Let $psi:X to X^{**}$ denote the canonical embedding of $X$ into its double dual $X^{**}$. We know that the weak$^*$ topology on $X^*$ is induced by the family of continuous linear functionals ${psi(x): xin X}$.

From this, can we say that every weak$^*$ continuous linear functional on $X^*$ is of the form $psi(x)$ for some $xin X$? If it is, I would like to know an explanation of the answer. If not then a counter example would help. I am a beginner on this domain and know very little. So, I do not know, if this question is trivial.

Thanks in advance.