Cone-saturated

In mathematics, specifically in order theory and functional analysis, if $C$ is a cone at 0 in a vector space $X$ such that $0\in C,$ then a subset $S\subseteq X$ is said to be $C$-saturated if $S=[S{]}_C,$ where $[S{]}_C:=(S+C)\cap (S-C).$ Given a subset $S\subseteq X,$ the $C$-saturated hull of $S$ is the smallest $C$-saturated subset of $X$ that contains $S.$Template:Sfn If $ℱ$ is a collection of subsets of $X$ then ${\left[ℱ\right]}_C:=\{[F{]}_C:F\in ℱ\}.$

If $𝒯$ is a collection of subsets of $X$ and if $ℱ$ is a subset of $𝒯$ then $ℱ$ is a fundamental subfamily of $𝒯$ if every $T\in 𝒯$ is contained as a subset of some element of $ℱ.$ If $𝒢$ is a family of subsets of a TVS $X$ then a cone $C$ in $X$ is called a $𝒢$-cone if $\{\overline{[G{]}_C}:G\in 𝒢\}$ is a fundamental subfamily of $𝒢$ and $C$ is a strict $𝒢$-cone if $\{[B{]}_C:B\in ℬ\}$ is a fundamental subfamily of $ℬ.$Template:Sfn

$C$-saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.

Properties

If $X$ is an ordered vector space with positive cone $C$ then $[S{]}_C=\bigcup \{[x,y]:x,y\in S\}.$Template:Sfn

The map $S\mapsto [S{]}_C$ is increasing; that is, if $R\subseteq S$ then $[R{]}_C\subseteq [S{]}_C.$ If $S$ is convex then so is $[S{]}_C.$ When $X$ is considered as a vector field over $ℝ,$ then if $S$ is balanced then so is $[S{]}_C.$Template:Sfn

If $ℱ$ is a filter base (resp. a filter) in $X$ then the same is true of ${\left[ℱ\right]}_C:=\{[F{]}_C:F\in ℱ\}.$

See also

• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link

Bibliography

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces