# How does one show that the complex conjugate of a function is dual to the function?

The convex conjugate

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

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