= Radial retraction onto a closed ball
{title2=$P_R(x)=x\min\{1,R/\|x\|\}$}
For $R>0$ in a <normed vector space>, define $P_R(0)=0$ and use the displayed formula elsewhere. This <continuous map> fixes the <closed ball> of radius $R$ and maps every point outside it to the boundary along its radial line. If a <continuous map> $T$ is defined only on that <closed ball>, $T\circ P_R$ is a continuous extension to the whole space. This extension is useful when applying a theorem whose hypotheses concern a globally defined <vector field>.
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