Let me introduce some definitions. Poly$(v_1, v_2,\ldots, v_k) = \{ x \in \mathbb{R}^n \ | \ (x, v_i) \geq 0 \ \ \forall i \}$ called a polyhedral cone. For any cone $C$, $ \ C^{\lor} = \{ x \in \mathbb{R}^n \ | \ \forall y \in C \ \ (x, y) \geq 0 \}$.
Let $C$ be a polyhedral cone. Prove that $C = (C^{\lor})^{\lor}$.
The only thing which I have managed to prove is that $C \subseteq (C^{\lor})^{\lor}$ for any cone $C$. So I have to prove that $(C^{\lor})^{\lor} \subseteq C$ for a polyhedral cone $C$, but I can't.