= Finite-dimensional interpolation form of Goldstine's theorem
{title2=$e(x)=x^{**}(e),\quad\|x\|<1+\varepsilon$}
For a finite-dimensional <vector subspace> $E\subseteq X^*$ and $x^{**}\in B_{X^{**}}$, exact interpolation on $E$ is possible by an $x\in X$ of <norm> less than $1+\varepsilon$. The restriction map $X\to E^*$ is onto and open. Separation of the image of the radius-$1+\varepsilon$ open ball would contradict $|x^{**}(e)|\leq\|e\|$. Scaling the interpolant slightly back into the closed <unit ball> proves the <Goldstine theorem>.
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