= Duality of copositive and completely positive cones
{title2=$\operatorname{COP}_n^*=\operatorname{CP}_n$}
Under the nonnegative-pairing convention for the <dual cone>, $\operatorname{CP}_n^*=\operatorname{COP}_n$ follows from $\langle A,xx^T\rangle_F=x^TAx$. Conversely, if $B\notin\operatorname{CP}_n$, <separation from a closed convex cone> gives a separating $A$ nonnegative on all generators and negative on $B$. That $A$ is a <copositive matrix>, so $B\notin\operatorname{COP}_n^*$. This proves the displayed equality using <closedness of the completely positive cone>.
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