= Approximable operator
{title2=$\overline{\{\text{bounded finite-rank operators}\}}^{\|\cdot\|}$}
A bounded <linear operator> between normed spaces is approximable if it is an <operator norm> limit of bounded <finite-rank operators>. If the target is a <Banach space>, every approximable operator is a <compact operator>. For a given tolerance, approximate the operator in <norm> by one <finite-rank operator>, cover that operator's bounded unit-ball image by finitely many small balls in its finite-dimensional range, and transfer the cover across the operator-norm error. This makes the original image <totally bounded>; its <closure> is <complete> in the Banach target and therefore <compact>. The converse on arbitrary <Banach spaces> would require an additional approximation hypothesis and is not part of this assertion.
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