 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 spaceHere 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 $E(x)+E^{\ast }(p)\ge ⟨x,p⟩$; the Fenchel-Moreau theorem recovers $E$ as $E^{\ast \ast }$ under the stated closed-convex hypotheses.