In mathematics, specifically in order theory and functional analysis, if is a cone at the origin in a topological vector space such that and if is the neighborhood filter at the origin, then is called normal if where and where for any subset is the -saturatation of
Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices.
If is a cone in a TVS then for any subset let be the -saturated hull of and for any collection of subsets of let If is a cone in a TVS then is normal if where is the neighborhood filter at the origin.
If is a collection of subsets of and if is a subset of then is a fundamental subfamily of if every is contained as a subset of some element of If is a family of subsets of a TVS then a cone in is called a -cone if is a fundamental subfamily of and is a strict -cone if is a fundamental subfamily of Let denote the family of all bounded subsets of
If is a cone in a TVS (over the real or complex numbers), then the following are equivalent: <ol> <li> is a normal cone.</li> <li> For every filter in if then </li> <li> There exists a neighborhood base in such that implies </ol> and if is a vector space over the reals then we may add to this list: <ol start=4> <li> There exists a neighborhood base at the origin consisting of convex, balanced, -saturated sets.</li> <li> There exists a generating family of semi-norms on such that for all and </li> </ol> and if is a locally convex space and if the dual cone of is denoted by then we may add to this list: <ol start=6> <li>For any equicontinuous subset there exists an equicontiuous such that </li> <li>The topology of is the topology of uniform convergence on the equicontinuous subsets of </li> </ol> and if is an infrabarreled locally convex space and if is the family of all strongly bounded subsets of then we may add to this list: <ol start=8> <li>The topology of is the topology of uniform convergence on strongly bounded subsets of </li> <li> is a -cone in
</li> <li> is a strict -cone in
</li> </ol> and if is an ordered locally convex TVS over the reals whose positive cone is then we may add to this list: <ol start=11> <li>there exists a Hausdorff locally compact topological space such that is isomorphic (as an ordered TVS) with a subspace of where is the space of all real-valued continuous functions on under the topology of compact convergence. </ol>
If is a locally convex TVS, is a cone in with dual cone and is a saturated family of weakly bounded subsets of then
If is a Banach space, is a closed cone in , and is the family of all bounded subsets of then the dual cone is normal in if and only if is a strict -cone.
If is a Banach space and is a cone in then the following are equivalent:
Suppose is an ordered topological vector space. That is, is a topological vector space, and we define whenever lies in the cone . The following statements are equivalent:
If the topology on is locally convex then the closure of a normal cone is a normal cone.
Suppose that is a family of locally convex TVSs and that is a cone in If is the locally convex direct sum then the cone is a normal cone in if and only if each is normal in
If is a locally convex space then the closure of a normal cone is a normal cone.
If is a cone in a locally convex TVS and if is the dual cone of then if and only if is weakly normal. Every normal cone in a locally convex TVS is weakly normal. In a normed space, a cone is normal if and only if it is weakly normal.
If and are ordered locally convex TVSs and if is a family of bounded subsets of then if the positive cone of is a -cone in and if the positive cone of is a normal cone in then the positive cone of is a normal cone for the -topology on