Legendre Transform is Involution

Theorem
The Legendre transform is an Involution.

Proof
Let $\map f x$ be a strictly convex real function.

Let $p = \map {f'} x$.

By definition of the Legendre transform, the transformed real function is of the form:


 * $\map {f^*} p=-\map f {\map x p}+p\map x p$

By Convexity of Function implies Convexity of its Legendre Transform, $f^*$ is strictly convex.

Let $t=\map { {f^*}'} p$.

Let $\map {\paren {f^*}^*} t=-\map {f^*} {\map p t}+t\map p t$.

Then:

$t$ is an arbitrary independent variable, hence can be renamed.

Set $t=x$.

Then:
 * $\tuple {t,f^{**} }=\tuple {x,f}$

which is the original pair of function and its variable.

Hence, by definition, the Legendre transform is an involution.