In mathematics, specifically in order theory and functional analysis, if
is a cone at the origin in a topological vector space
is the neighborhood filter at the origin, then
[1] Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices.
is a cone in a TVS
is the neighborhood filter at the origin.
is contained as a subset of some element of
is a family of subsets of a TVS
denote the family of all bounded subsets of
(over the real or complex numbers), then the following are equivalent:[1] and if
is a vector space over the reals then we may add to this list:[1] and if
is a locally convex space and if the dual cone of
is an infrabarreled locally convex space and if
is the family of all strongly bounded subsets of
is an ordered locally convex TVS over the reals whose positive cone is
is a locally convex TVS,
is a saturated family of weakly bounded subsets of
is the family of all bounded subsets of
is an ordered topological vector space.
is a topological vector space, and we define
The following statements are equivalent:[3] If the topology on
is locally convex then the closure of a normal cone is a normal cone.
is a family of locally convex TVSs and that
is the locally convex direct sum then the cone
is a locally convex space then the closure of a normal cone is a normal cone.
is a cone in a locally convex TVS
[1] Every normal cone in a locally convex TVS is weakly normal.
[1] In a normed space, a cone is normal if and only if it is weakly normal.
are ordered locally convex TVSs and if
is a family of bounded subsets of