Dieudonné's theorem

In mathematics, Dieudonné's theorem, named after Jean Dieudonné, is a theorem on when the Minkowski sum of closed sets is closed.

Statement

Let [math]\displaystyle{ X }[/math] be a locally convex space and [math]\displaystyle{ A,B \subset X }[/math] nonempty closed convex sets. If either [math]\displaystyle{ A }[/math] or [math]\displaystyle{ B }[/math] is locally compact and [math]\displaystyle{ \operatorname{recc}(A) \cap \operatorname{recc}(B) }[/math] (where [math]\displaystyle{ \operatorname{recc} }[/math] gives the recession cone) is a linear subspace, then [math]\displaystyle{ A - B }[/math] is closed.