Normal cone (functional analysis)
In mathematics, specifically in order theory and functional analysis, if formula_1 is a cone at the origin in a topological vector space formula_2 such that formula_3 and if formula_4 is the neighborhood filter at the origin, then formula_1 is called normal if formula_6 where formula_7 and where for any subset formula_8 formula_9 is the formula_1-saturatation of formula_11 Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices. Characterizations. If formula_1 is a cone in a TVS formula_2 then for any subset formula_14 let formula_15 be the formula_1 -saturated hull of formula_14 and for any collection formula_18 of subsets of formula_2 let formula_20 If formula_1 is a cone in a TVS formula_2 then formula_1 is normal if formula_6 where formula_4 is the neighborhood filter at the origin. If formula_26 is a collection of subsets of formula_2 and if formula_28 is a subset of formula_26 then formula_28 is a fundamental subfamily of formula_26 if every formula_32 is contained as a subset of some element of formula_33 If formula_34 is a family of subsets of a TVS formula_2 then a cone formula_1 in formula_2 is called a formula_34-cone if  formula_39 is a fundamental subfamily of formula_34 and formula_1 is a strict formula_34-cone if formula_43 is a fundamental subfamily of formula_44 Let formula_45 denote the family of all bounded subsets of formula_46 If formula_1 is a cone in a TVS formula_2 (over the real or complex numbers), then the following are equivalent: and if formula_2 is an ordered locally convex TVS over the reals whose positive cone is formula_50 then we may add to this list: If formula_2 is a locally convex TVS, formula_1 is a cone in formula_2 with dual cone formula_54 and formula_34 is a saturated family  of weakly bounded subsets of formula_56 then If formula_2 is a Banach space, formula_1 is a closed cone in formula_71, and formula_72 is the family of all bounded subsets of formula_73 then the dual cone formula_57 is normal in formula_73 if and only if formula_1 is a strict formula_45-cone. If formula_2 is a Banach space and formula_1 is a cone in formula_2 then the following are equivalent: Sufficient conditions. If the topology on formula_2 is locally convex then the closure of a normal cone is a normal cone. Suppose that formula_113 is a family of locally convex TVSs and  that formula_114 is a cone in formula_115 If formula_116 is the locally convex direct sum then the cone formula_117 is a normal cone in formula_2 if and only if each formula_119 is normal in formula_115 If formula_2 is a locally convex space then the closure of a normal cone is a normal cone. If formula_1 is a cone in a locally convex TVS formula_2 and if formula_57 is the dual cone of formula_50 then formula_126 if and only if formula_1 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 formula_2 and formula_96 are ordered locally convex TVSs and if formula_34 is a family of bounded subsets of formula_71 then if the positive cone of formula_2 is a formula_34-cone in formula_2 and if the positive cone of formula_96 is a normal cone in formula_96 then the positive cone of formula_111 is a normal cone for the formula_34-topology on formula_139