View source for Weak order unit

{{one source|date=July 2020}}

In mathematics, specifically in [[order theory]] and [[functional analysis]], an element <math>x</math> of a [[vector lattice]] <math>X</math> is called a '''weak order unit''' in <math>X</math> if <math>x \geq 0</math> and also for all <math>y \in X,</math> <math>\inf \{ x, |y| \} = 0 \text{ implies } y = 0.</math>{{sfn|Schaefer|Wolff|1999|pp=234–242}} 

== Examples ==

* If <math>X</math> is a [[Separable space|separable]] [[Fréchet space|Fréchet]] [[topological vector lattice]] then the set of weak order units is dense in the positive cone of <math>X.</math>{{sfn|Schaefer|Wolff|1999|pp=204–214}}

== See also ==

* {{annotated link|Quasi-interior point}}
* {{annotated link|Vector lattice}}

== Citations ==

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==References==

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->

{{Ordered topological vector spaces}}
{{Functional analysis}}

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[[Category:Functional analysis]]