I have a locally convex space $X$ with topological dual $X^*$ and coupling $\langle x,x^* \rangle:=x^*(x),\ x\in X,\ x^*\in X^*$.
For $f:X\to\overline{\mathbb{R}}$ one defines its convex conjugate by $f^*(x^*):=\sup\{\langle x,x^* \rangle-f(x)\mid x\in X\}$ and similarly for $g:X^*\to\overline{\mathbb{R}}$ one defines its convex conjugate by $g^*(x):=\sup\{\langle x,x^* \rangle-g(x^*)\mid x^*\in X^*\}$.
The biconjugate formula states that for a proper convex lower semicontinuous $f:X\to\overline{\mathbb{R}}$ one has $f^{**}=f$.
The proof we went over in class is based on $X$ being Hausdorff separated and relies on separation theorems for the epigraph.
My question is whether the biconjugate formula holds when the space $X$ is NOT Hausdorff separated?