# Proof of properties of dual cone

Show that if C $\subseteq$ D then $D^*$ $\subseteq$ $C^*$

where * is dual cone operation. Can somebody explain it.

#### Solutions Collecting From Web of "Proof of properties of dual cone"

In view of the comment, I imagine that $C$ and $D$ are subsets of some Euclidean space $E$, that $C^*=\{x\in E:(\forall y\in C)\,x\cdot y\geq0\}$, and similarly for $D^*$. If so, and if $C\subseteq D$, then any $x\in D^*$ satisfies $x\cdot y\geq 0$ for all $y\in D$, and that includes all $y\in C$, so $x\in C^*$. Since $x$ was an arbitrary element of $D^*$, that proves $D^*\subseteq C^*$.