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 ∈ C, then a subset S of X is said to be C-saturated if S = [S]C, where [S]C := (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.[1] If is a collection of subsets of X in X then .

If is a collection of subsets of X 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 X then a cone C in X is called a -cone if is a fundamental subfamily of and C is a strict -cone if is a fundamental subfamily of .[1]

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 .[1]

The map is increasing (i.e. if RS then [R]C ⊆ [S]C). If S is convex then so is [S]C. When X is considered as a vector field over , then if S is balanced then so is [S]C.[1]

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

See also

References

  1. Schaefer & Wolff 1999, pp. 215–222.
  • 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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