Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones,


$A^\ast=C,$


$B^\ast=C,$


can we state that $A=B$? Is the dual cone of a cone is unique? 


the definition of dual cone here is: 


The dual cone C\* of a subset C in a linear space X, e.g. Euclidean space $R^n$, with topological dual space X\* is the set


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



Is dual cone unique?
The answer to the edited question is yes. For any non-empty, closed, convex cone $C\subseteq V$ for any locally convex and hausdorff topological vector space $V$ the equation $C^{\ast\ast}=C$ holds. One inclusion follows immediately from the definition, the other follows easily by aiming for a contradiction and applying the Hahn-Banach theorem about separation of closed convex sets from points.


