= Solution
For a real <Banach space> $X$ and a <proper convex function> $E:X\to\mathbb R\cup\{+\infty\}$, the <convex conjugate> is the extended-valued function on the <continuous dual space>
$$
\boxed{E^*(p)=\sup_{x\in X}\{\langle x,p\rangle-E(x)\},\qquad p\in X^*.}
$$
Here proper means that $E$ never takes $-\infty$ and is finite somewhere. The pairing is the duality pairing, or the inner product after identification in a <Hilbert space>. The conjugate is convex and lower semicontinuous because it is a supremum of continuous affine functions. Its definition gives the <Fenchel–Young inequality> $E(x)+E^*(p)\geq\langle x,p\rangle$; the <Fenchel-Moreau theorem> recovers $E$ as $E^{**}$ under the stated closed-convex hypotheses.
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