The image of non-reflexive Banach space in its bidual is at large distance from some unit vector in the bidual I am having some difficulties with part of a problem, I am working on. > Let $X$ be a non-reflexive Banach space and let $i: X \to X^{**}$ be the canonical embedding. Show that for given $\epsilon > 0$ there exists some $x^{**} \in X^{**}$, such that $\|x^{**}\|=1$ and $$\inf_{x \in X} \|x^{**} - i(x)\|_{X^{**}} > 1- \epsilon.$$ I have no idea how to show more than the existence of some $x^{**} \in X^{**}$, such that $\|x^{**}\|=1$ and $$\inf_{x \in X} \|x^{**} - i(x)\|_{X^{**}} > 0,$$ which is by definition of reflexivity and closeness of $i(X)$.

Simply use the Riesz lemma as $i(X)$ is norm-closed in $X^{**}$.