Let S1, S2 be nonempty sets in a normed linear space.

Whatever the nature of the set S, S* and S** are always cones (in X* and X respectively) and S Š† S**. Under what conditions is S identical to its bipolar S** (rather than a proper subset)? Clearly, a necessary condition is that S is a convex cone. The additional requirement is that S is closed. This is another implication of the strong separation theorem.

SI s S2 = S; s S