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The convex conjugate

$${\displaystyle f^{\star }:X^{*}\to \mathbb {R} \cup \{+\infty \}} $$ is defined in terms of the supremum by

$$f^{\star }\left(x^{*}\right):=\sup \left\{\left.\left\langle x^{*},x\right\rangle -f\left(x\right)\right|x\in X\right\}$$

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Well, this proof is apparently called the Fenchel-Moreau Theorem and you can find its proof here: https://www2.math.ethz.ch/education/bachelor/lectures/hs2015/math/mf/lecture7notes

I must say I don't understand the proof at all yet.

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