= Quotient-distance criterion for compactness
{title2=$T\text{ compact}\iff\inf_{\dim Y<\infty}\|\pi_YT\|=0$}
For a <bounded linear operator> into a <Banach space>, compactness is equivalent to the displayed condition, where $\pi_Y$ is the <quotient map> by a finite-dimensional subspace of the target. A finite net for the image of the unit ball gives such a subspace by taking its span. Conversely, if the image lies within $\varepsilon$ of a finite-dimensional subspace, choose approximants there; they remain bounded because the original image is bounded. A finite net for these approximants gives a slightly larger finite net for the image. This proves <total boundedness> and hence compactness. This condition does not require choosing the approximants by a <linear map>, and therefore does not assume an approximation property for the <Banach space>.
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