= Solution
For a <functional> $E$ on a real <Banach space>, its <convex conjugate> is defined on the <dual space> by
$$
\boxed{E^*(p)=\sup_{u\in\mathcal U}\{\langle p,u\rangle-E(u)\},\qquad p\in\mathcal U^*.}
$$
Here $\langle p,u\rangle$ is the <dual pairing>. For complex spaces, use its real part, equivalently regard the space as real for <convex analysis>. The <convex conjugate> is a <convex function>, being the supremum of <affine functions> of $p$, even when $E$ itself is not <convex>.
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