Dual set of the unit ball is part of the unit ball. Define the unit ball centered at the origin as $B=\{x\in\mathbb{R}^d\mid \|x\|\leq 1\}$.
Define the dual set of set $X$ as $X^*=\{y\in\mathbb{R}^d\mid\langle x,y \rangle\leq 1\ \forall x\in X\}$.
I'm attempting to prove that for any set $C\subseteq\mathbb{R}^d$ it holds that $C=B$ if and only if $C=C^*$. I've managed to prove the implication from the right to the left by the Cauchy-Schwarz inequality.
However, I'm having some trouble with the converse implication. In particular, how to prove that $C=B$ implies $C^*\subseteq C$?

Suppose $C=C^*$, you have $<x,x>=\|x\|\leq 1$ since $x\in C,x\in C^*$ thus $C\subset B$. This implies $B^*=B\subset C^*=C$ since it is easily shown that $X\subset Y$ implies $Y^*\subset X^*$. done.
