Weak order unit

In mathematics, specifically in order theory and functional analysis, an element $x$ of a vector lattice $X$ is called a weak order unit in $X$ if $x \geq 0$ and also for all $y \in X,$ $inf {x, | y |} = 0 implies y = 0.$

Examples

If $X$ is a separable Fréchet topological vector lattice then the set of weak order units is dense in the positive cone of $X .$

Citations

↑ Schaefer & Wolff 1999, pp. 234–242.
↑ Schaefer & Wolff 1999, pp. 204–214.

References

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.

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[FOOTNOTESchaeferWolff1999234–242-1] Schaefer & Wolff 1999, pp. 234–242.

[FOOTNOTESchaeferWolff1999204–214-2] Schaefer & Wolff 1999, pp. 204–214.

v t e Ordered topological vector spaces
Basic concepts	Ordered vector space Partially ordered space Riesz space Order topology Order unit Positive linear operator Topological vector lattice Vector lattice
Types of orders/spaces	AL-space AM-space Archimedean Banach lattice Fréchet lattice Locally convex vector lattice Normed lattice Order bound dual Order dual Order complete Regularly ordered
Types of elements/subsets	Band Cone-saturated Lattice disjoint Dual/Polar cone Normal cone Order complete Order summable Order unit Quasi-interior point Solid set Weak order unit
Topologies/Convergence	Order convergence Order topology
Operators	Positive State
Main results	Freudenthal spectral

Weak order unit

Examples

See also

Citations

References