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In [[order theory]] and [[functional analysis]], if ''C'' is a cone at 0 in a vector space ''X'' such that 0 ∈ ''C'', then a subset ''S'' of ''X'' is said to be '''''C''-saturated''' if ''S'' = [''S'']<sub>C</sub>, where [''S'']<sub>C</sub> := (S + C) ∩ (S - C).
Given a subset ''S'' of ''X'', the '''''C''-saturated hull''' of ''S'' is the smallest ''C''-saturated subset of ''X'' that contains ''S''.
''C''-saturated sets play an important role in the theory of [[ordered topological vector space]]s and [[topological vector lattice]]s.
== Properties ==
If ''X'' is an ordered vector space then [''S'']<sub>C</sub> = <math>\cup \left\{ [x, y] : x, y \in S \right\}</math>.
The map <math>S \mapsto [S]_C</math> is increasing (i.e. if ''R'' ⊆ ''S'' then [''R'']<sub>C</sub> ⊆ [''S'']<sub>C</sub>).
If ''S'' is convex then so is [''S'']<sub>C</sub>.
When ''X'' is considered as a vector field over <math>\mathbb{R}</math>, then if ''S'' is [[balanced set|balanced]] then so is [''S'']<sub>C</sub>.
If <math>\mathcal{F}</math> is a [[filter base]] (resp. a filter) in ''X'' then the same is true of <math>\left[ \mathcal{F} \right]_{C} := \left\{ [ F ]_{C} : F \in \mathcal{F} \right\}</math>.
== See also ==
* [[Banach lattice]]
* [[Fréchet lattice]]
* [[Locally convex vector lattice]]
* [[Vector lattice]]
== References ==
* <!-- -->
<!--- Categories --->
[[Category:Functional analysis]]
Given a subset ''S'' of ''X'', the '''''C''-saturated hull''' of ''S'' is the smallest ''C''-saturated subset of ''X'' that contains ''S''.
''C''-saturated sets play an important role in the theory of [[ordered topological vector space]]s and [[topological vector lattice]]s.
== Properties ==
If ''X'' is an ordered vector space then [''S'']<sub>C</sub> = <math>\cup \left\{ [x, y] : x, y \in S \right\}</math>.
The map <math>S \mapsto [S]_C</math> is increasing (i.e. if ''R'' ⊆ ''S'' then [''R'']<sub>C</sub> ⊆ [''S'']<sub>C</sub>).
If ''S'' is convex then so is [''S'']<sub>C</sub>.
When ''X'' is considered as a vector field over <math>\mathbb{R}</math>, then if ''S'' is [[balanced set|balanced]] then so is [''S'']<sub>C</sub>.
If <math>\mathcal{F}</math> is a [[filter base]] (resp. a filter) in ''X'' then the same is true of <math>\left[ \mathcal{F} \right]_{C} := \left\{ [ F ]_{C} : F \in \mathcal{F} \right\}</math>.
== See also ==
* [[Banach lattice]]
* [[Fréchet lattice]]
* [[Locally convex vector lattice]]
* [[Vector lattice]]
== References ==
* <!-- -->
<!--- Categories --->
[[Category:Functional analysis]]
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