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In mathematics, specifically in order theory and functional analysis, if C {\displaystyle C} is a cone at the origin in a topological vector space X {\displaystyle X} such that 0 ∈ C {\displaystyle 0\in C} and if U {\displaystyle {\mathcal {U}}} is the neighborhood filter at the origin, then C {\displaystyle C} is called normal if U = C , {\displaystyle {\mathcal {U}}=\left_{C},} where C := { C : U ∈ U } {\displaystyle \left_{C}:=\left\{_{C}:U\in {\mathcal {U}}\right\}} and where for any subset S ⊆ X , {\displaystyle S\subseteq X,} C := ∩ {\displaystyle _{C}:=\cap } is the C {\displaystyle C} -saturatation of S . {\displaystyle S.}
Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices.