Can we show the closed unit ball of $\hat{X}$ is norm closed in $X^{**}$?

Question

Definition: Let $X$ be a topological vector space and let $x\in X$. Then $x$ defines a linear functional $\hat{x}$ on $X^*$ via $\hat{x}(f)=f(x)$ $(f\in X^*)$.

Let $X$ be a normed space. Then $\hat{X}\subseteq X^{**}$. Let $B_\hat{X}$ be the closed unit ball of $\hat{X}$. Then how to show $B_\hat{X}$ is norm closed in $X^{**}$?

Thank you in advance!!

Answer 1

If $X$ is merely a normed space, $B_{\hat X}$ might fail to be closed. In fact, $B_{\hat X}$ is closed if and only if $X$ is complete.