Suppose $X$ is a Banach Space and $X^{**}$ is the bidual, $J:X\to X^{**}$ is the canonical map from $X$ to $X^{**}$. It's well known that $J$ might not be surjective.
My question is that: given $\phi \in X^{**}$, can we find a sequence $\{x_n\}\subset X$ such that $\{J(x_n)\}$ converges to $\phi$ in some sense (strongly or weakly)?
It is not dense in the norm unless $X$ is reflexive but it is always dense in the weak* topology of $X^{**}$.
[If $X$ is not weak* dense in $X^{**}$ there would be an $x^{*} \in X^{*}$ other than $0$ which vanishes on $X$. But this is impossible. I am using Hahn Banach Theorem and the fact that the dual of $(X^{**},weak*)$ is $X^{*}$].
As pointed out by KeeoerOfSecrets below sequences cannot be use for denseness in weak* topology.
