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 = C, where 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 $S.$
If $\mathcal$ is a collection of subsets of $X$ then \left \mathcal \rightC := \left\.

If $\mathcal$ is a collection of subsets of $X$ and if $\mathcal$ is a subset of $\mathcal$ then $\mathcal$ is a fundamental subfamily of $\mathcal$ if every $T \in \mathcal$ is contained as a subset of some element of $\mathcal.$
If $\mathcal$ is a family of subsets of a TVS $X$ then a cone $C$ in $X$ is called a $\mathcal$-cone if $\left\$ is a fundamental subfamily of $\mathcal$ and $C$ is a strict $\mathcal$-cone if $\left\$ is a fundamental subfamily of $\mathcal.$

$C$-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 C = \bigcup \left\.

The map S \mapsto C is increasing; that is, if $R \subseteq S$ then C \subseteq C.
If $S$ is convex then so is C. When $X$ is considered as a vector field over $\R,$ then if $S$ is balanced then so is C.

If $\mathcal$ is a filter base (resp. a filter) in $X$ then the same is true of \left \mathcal \rightC := \left\.

See also

*
*
*
*

References

Bibliography

*
*

{{Ordered topological vector spaces

Functional analysis