Theorem 21.11.24258227E5D497

Proof. For proper $F$, all assertions are the finite-dimensional Fenchel theorem, whose separation proof is given earlier: $F^*$ is closed and proper, $(\operatorname{cl}F)^*=F^*$, and $F^{**}=\operatorname{cl}F$.

For completeness consider improper convex functions. If $F\equiv+\infty$, then $F^*\equiv-\infty$ and $F^{**}\equiv+\infty$. Otherwise an improper $F$ takes $-\infty$ somewhere, so every $F^*(p)$ is $+\infty$, its biconjugate is identically $-\infty$, and this is exactly the closure prescribed in the definition. Its conjugate is again identically $+\infty$. Neither conjugate is proper in these improper cases. This proves the equivalence of properness and all the identities also at the extended-valued extremes.