Let $X$ be a Hibert space, $x \mapsto \|x\| := \sqrt{x^Tx}$ be the euclidean norm on $X$, and $f:X \rightarrow (-\infty,+\infty]$ an extended real-valued function. Define the convex conjugate of $f$, denoted $f^*$, by
$$f^*(y) := \sup_{x \in X}x^Ty - f(x), \; \forall y \in X.$$
Note that $f^*$ is always convex l.s.c (being the supremum of affine functions) without any assumptions whatsoever on $f$.
Question: When do we have $f^* = f$ ? Checkout the answer here.
