In Folland, Exercise 5.52(c), the question is to show that the relative topology on $X$ induced by the weak-$*$ topology on $X^{**}$ is the weak topology on $X$.
It is not clear to me what is meant by weak-$*$ topology on $X^{**}$. If I understand it correctly, weak-$*$ topology is the coarsest topology which makes the evaluation functionals continuous. So the weak-$*$ topology should apply to either $X^{*}$ or $X^{***}$, not $X^{**}$.
Thanks!
The double dual $X^{**}$ is the dual of $X^*$, so of course it has a weak$^*$-topology: it is the one given by evaluation on the elements of $X^*$.
So, if $\{\alpha_j\}\subset X^{**}$ converges to zero weak$^*$, this means that for every $f\in X^*$, $\alpha_j(f)\to0$.
When we consider the canonical embedding $X\subset X^{**}$, if we have a net $\{x_j\}\subset X$, then $x_j\to0$ in the relative weak$^*$-topology of $X^{**}$ if, for every $f\in X^*$, we have $$\hat x(f)\to0.$$ But the embedding consists precisely on defining $\hat x(f)=f(x)$. So $f(x_j)\to0$ for all $f\in X^*$, which is precisely the statement that $x_j\to0$ weakly.
