Every element of $J(X) \subset X^{**}$ attains their norm

Question

Let $X$ be a Banach space and $X^*$, $X^{**}$ are the dual and bi-dual of $X$ respectively. Let $J:X \to X^{**}$ be the canonical embedding of $X$ into $X^{**}$. Let $\phi \in J(X)$. Then I want to show that, there exists an element $x^*\in X^*$ with $\|x^*\|=1$ such that $\vert \phi(x^*)\vert= \|\phi\|.$ I guess I have to use Hahn-Banach theorem, but I am unable to figure this out. Please help me to solve this. Thank you for your time and help.

Answer 1

Let $\phi=J(x).$ By the Hahn-Banach theorem, $\|\phi\|=\|x\|$ and there exist an $x^*\in X^*$ such that $\|x^*\|=1$ and $x^*(x)=\|x\|,$ i.e. $\phi(x^*)=\|\phi\|.$