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 $y\in X,$ $\inf\{x,|y|\}=0{\text{ implies }}y=0.$ [1]

Examples

If $X$ is a separable Fréchet topological vector lattice then the set of weak order units is dense in the positive cone of $X.$ [2]

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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

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

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

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