Equivalence of defining the Fenchel conjugate as $\sup_x\{\langle x^*, x \rangle - f(x)\}$ or $\inf_x\{f(x)-\langle x^*, x \rangle\}$

Question

The canonical definition of Fenchel conjugate function is that for a function $f(x)$, $$f^*(x^*) = \textsf{sup}_{x}\{\langle x^*, x \rangle - f(x)\}$$ which is equivalent to $f^*(x^*) =-\textsf{inf}_x\{f(x)-\langle x^*, x \rangle\}$. Recently as I was reading Optimization by vector space methods, I noticed that in Section 8.6, in order to prove the Lagrange duality theorem, the book introduces another dual functional: $$\textsf{inf}_{x}\{f(x)+ \langle x^*, x\rangle\}$$ The book claims that those two definitions are essentially the same. I can see that they have the same geometric interpretation, but I failed to prove that they are the same rigorously.

Answer 1

If you define $$f^*(y) = \textsf{sup}_{x}\{\langle x, y\rangle - f(x)\}$$ $$g(y) = \textsf{inf}_{x}\{f(x)+ \langle x, y\rangle\},$$ then $f^*(y) = -g(-y)$.