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We know that for a vector space $V$ there is a natural map $\iota\colon V\to V^{**}$ sending $v$ to $v^{**}$.

When $V$ is a normed space, $\iota$ is an isometry, however it may not be surjective. My question is whether arbitrary vectors of $V^{**}$ can be approximated (in any sense) by the vectors in the image of $\iota$ for an arbitrary Banach space $V$.

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Yes, this is Goldstine's theorem which asserts that $\iota(V)$ is weakly* dense in $V^{**}$. Of course, $\iota(V)$ is norm-closed in $V^{**}$ as $\iota$ is an isometry.

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