In mathematics, specifically in order theory and functional analysis, if ${\displaystyle C}$ is a cone at the origin in a topological vector space ${\displaystyle X}$ such that ${\displaystyle 0\in C}$ and if ${\displaystyle {\mathcal {U}}}$ is the neighborhood filter at the origin, then ${\displaystyle C}$ is called normal if ${\displaystyle {\mathcal {U}}=\left[{\mathcal {U}}\right]_{C},}$ where ${\displaystyle \left[{\mathcal {U}}\right]_{C}:=\left\{[U]_{C}:U\in {\mathcal {U}}\right\}}$ and where for any subset ${\displaystyle S\subseteq X,}$ ${\displaystyle [S]_{C}:=(S+C)\cap (S-C)}$ is the ${\displaystyle C}$-saturatation of ${\displaystyle S.}$

Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices.

If ${\displaystyle C}$ is a cone in a TVS ${\displaystyle X}$ then for any subset ${\displaystyle S\subseteq X}$ let ${\displaystyle [S]_{C}:=\left(S+C\right)\cap \left(S-C\right)}$ be the ${\displaystyle C}$-saturated hull of ${\displaystyle S\subseteq X}$ and for any collection ${\displaystyle {\mathcal {S}}}$ of subsets of ${\displaystyle X}$ let ${\displaystyle \left[{\mathcal {S}}\right]_{C}:=\left\{\left[S\right]_{C}:S\in {\mathcal {S}}\right\}.}$ If ${\displaystyle C}$ is a cone in a TVS ${\displaystyle X}$ then ${\displaystyle C}$ is normal if ${\displaystyle {\mathcal {U}}=\left[{\mathcal {U}}\right]_{C},}$ where ${\displaystyle {\mathcal {U}}}$ is the neighborhood filter at the origin.

If ${\displaystyle {\mathcal {T}}}$ is a collection of subsets of ${\displaystyle X}$ and if ${\displaystyle {\mathcal {F}}}$ is a subset of ${\displaystyle {\mathcal {T}}}$ then ${\displaystyle {\mathcal {F}}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {T}}}$ if every ${\displaystyle T\in {\mathcal {T}}}$ is contained as a subset of some element of ${\displaystyle {\mathcal {F}}.}$ If ${\displaystyle {\mathcal {G}}}$ is a family of subsets of a TVS ${\displaystyle X}$ then a cone ${\displaystyle C}$ in ${\displaystyle X}$ is called a ${\displaystyle {\mathcal {G}}}$-cone if ${\displaystyle \left\{{\overline {\left[G\right]_{C}}}:G\in {\mathcal {G}}\right\}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {G}}}$ and ${\displaystyle C}$ is a strict ${\displaystyle {\mathcal {G}}}$-cone if ${\displaystyle \left\{\left[G\right]_{C}:G\in {\mathcal {G}}\right\}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {G}}.}$ Let ${\displaystyle {\mathcal {B}}}$ denote the family of all bounded subsets of ${\displaystyle X.}$

If ${\displaystyle C}$ is a cone in a TVS ${\displaystyle X}$ (over the real or complex numbers), then the following are equivalent:

and if ${\displaystyle X}$ is a vector space over the reals then we may add to this list:

and if ${\displaystyle X}$ is a locally convex space and if the dual cone of ${\displaystyle C}$ is denoted by ${\displaystyle X^{\prime }}$ then we may add to this list:

and if ${\displaystyle X}$ is an infrabarreled locally convex space and if ${\displaystyle {\mathcal {B}}^{\prime }}$ is the family of all strongly bounded subsets of ${\displaystyle X^{\prime }}$ then we may add to this list: