Weak order unit
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In mathematics, specifically in order theory and functional analysis, an element [math]\displaystyle{ x }[/math] of a vector lattice [math]\displaystyle{ X }[/math] is called a weak order unit in [math]\displaystyle{ X }[/math] if [math]\displaystyle{ x \geq 0 }[/math] and also for all [math]\displaystyle{ y \in X, }[/math] [math]\displaystyle{ \inf \{ x, |y| \} = 0 \text{ implies } y = 0. }[/math][1]
Examples
- If [math]\displaystyle{ X }[/math] is a separable Fréchet topological vector lattice then the set of weak order units is dense in the positive cone of [math]\displaystyle{ X. }[/math][2]
See also
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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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