dual cone is closed

Question

I have been trying to understand why dual of a cone is closed, no matter the cone is closed or not. I know the proof is 'It is because dual cone is an intersection of closed halfspaces.'. I just do not understand how it is linked to the definition of the dual cone.

Answer 1

Dua cone of $C$ is defined to be

$$C^*=\left\{ y \in X^*: \langle y,x\rangle \geq 0 , \forall x \in C \right\}$$

For each $x$, $\left\{y :\langle y,x \rangle \geq 0\right\}$ is a closed half space. Hence $C^*$ is an intersection of closed halfspaces.