In mathematics, specifically in order theory and functional analysis, if is a cone at 0 in a vector space such that then a subset is said to be -saturated if where Given a subset the -saturated hull of is the smallest -saturated subset of that contains [1] If is a collection of subsets of then

If is a collection of subsets of and if is a subset of then is a fundamental subfamily of if every is contained as a subset of some element of If is a family of subsets of a TVS then a cone in is called a -cone if is a fundamental subfamily of and is a strict -cone if is a fundamental subfamily of [1]

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

Properties

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If   is an ordered vector space with positive cone   then  [1]

The map   is increasing; that is, if   then   If   is convex then so is   When   is considered as a vector field over   then if   is balanced then so is  [1]

If   is a filter base (resp. a filter) in   then the same is true of  

See also

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References

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  1. ^ a b c d Schaefer & Wolff 1999, pp. 215–222.

Bibliography

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  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.