Legendre Transform is Involution

Theorem
Legendre Transform is Involution,

Proof
Let Let $f \left({x}\right)$ be a strictly convex real function.

Let $p= f' \left({x}\right)$.

By the definition of Legendre Transform, the transformed real function is of the form


 * $f^* \left({p}\right) = -f \left({x \left({p}\right)}\right) + p x \left({p} \right)$

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

Let $ t= { f^*}' \left({ p } \right)$.

Let $\left( { f^* } \right)^* \left({ t } \right)= -f^* \left({ p \left({ t } \right) } \right)+t p \left({ t } \right)$.

Then:

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

Set $t=x$.

Then $\left({ t, f^{**} } \right)=\left({ x, f } \right)$, which is the original pair of function and its variable.

By the definition of involution, the Legendre Transform is an involution.