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 ${\displaystyle {\mathcal {F}}}$ is a collection of subsets of X in X then ${\displaystyle \left[{\mathcal {F}}\right]_{C}:=\left\{[F]_{C}:F\in {\mathcal {F}}\right\}}$.

If ${\displaystyle {\mathcal {T}}}$ is a collection of subsets of X and if ${\displaystyle {\mathcal {F}}}$ is a subset of ${\displaystyle {\mathcal {T}}}$ then ${\displaystyle {\mathcal {F}}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {T}}}$ if every ${\displaystyle T\in {\mathcal {T}}}$ is contained as a subset of some element of ${\displaystyle {\mathcal {F}}}$. If ${\displaystyle {\mathcal {G}}}$ is a family of subsets of a TVS X then a cone C in X is called a ${\displaystyle {\mathcal {G}}}$-cone if ${\displaystyle \left\{{\overline {\left[G\right]_{C}}}:G\in {\mathcal {G}}\right\}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {G}}}$ and C is a strict ${\displaystyle {\mathcal {G}}}$-cone if ${\displaystyle \left\{\left[B\right]_{C}:B\in {\mathcal {B}}\right\}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {B}}}$.[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 ${\displaystyle [S]_{C}=\bigcup \left\{[x,y]:x,y\in S\right\}}$.[1]

The map ${\displaystyle S\mapsto [S]_{C}}$ is increasing (i.e. if R ⊆ S then [R]_C ⊆ [S]_C). If S is convex then so is [S]_C. When X is considered as a vector field over ${\displaystyle \mathbb {R} }$, then if S is balanced then so is [S]_C.[1]

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

See also

• Banach lattice
• Fréchet lattice
• Locally convex vector lattice
• Vector lattice

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.