polar we have For any finite collection of sets If then and An immediate corollary is that ; equality necessarily holds when is finite and may fail to hold if is infinite. and If is a cone in then If is a family of -closed subsets of containing then the real polar of is the closed convex hull of If then For a closed convex cone in a real vector space the polar cone is the polar of ; that is, where See also Notes References Bibliography Functional analysis Topological vector spaces