Order convergence

In mathematics, specifically in order theory and functional analysis, a filter [math]\displaystyle{ \mathcal{F} }[/math] in an order complete vector lattice [math]\displaystyle{ X }[/math] is order convergent if it contains an order bounded subset (that is, is contained in an interval of the form [math]\displaystyle{ [a, b] := \{ x \in X : a \leq x \text{ and } x \leq b \} }[/math]) and if [math]\displaystyle{ \mathcal{F}, }[/math] [math]\displaystyle{ \sup \left\{ \inf S : S \in \operatorname{OBound}(X) \cap \mathcal{F} \right\} = \inf \left\{ \sup S : S \in \operatorname{OBound}(X) \cap \mathcal{F} \right\}, }[/math] where [math]\displaystyle{ \operatorname{OBound}(X) }[/math] is the set of all order bounded subsets of X, in which case this common value is called the order limit of [math]\displaystyle{ \mathcal{F} }[/math] in [math]\displaystyle{ X. }[/math]

Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology.

Definition

A net [math]\displaystyle{ \left(x_{\alpha}\right)_{\alpha \in A} }[/math] in a vector lattice [math]\displaystyle{ X }[/math] is said to decrease to [math]\displaystyle{ x_0 \in X }[/math] if [math]\displaystyle{ \alpha \leq \beta }[/math] implies [math]\displaystyle{ x_{\beta} \leq x_{\alpha} }[/math] and [math]\displaystyle{ x_0 = inf \left\{ x_{\alpha} : \alpha \in A \right\} }[/math] in [math]\displaystyle{ X. }[/math] A net [math]\displaystyle{ \left(x_{\alpha}\right)_{\alpha \in A} }[/math] in a vector lattice [math]\displaystyle{ X }[/math] is said to order-converge to [math]\displaystyle{ x_0 \in X }[/math] if there is a net [math]\displaystyle{ \left(y_{\alpha}\right)_{\alpha \in A} }[/math] in [math]\displaystyle{ X }[/math] that decreases to [math]\displaystyle{ 0 }[/math] and satisfies [math]\displaystyle{ \left|x_{\alpha} - x_0\right| \leq y_{\alpha} }[/math] for all [math]\displaystyle{ \alpha \in A }[/math].

Order continuity

A linear map [math]\displaystyle{ T : X \to Y }[/math] between vector lattices is said to be order continuous if whenever [math]\displaystyle{ \left(x_{\alpha}\right)_{\alpha \in A} }[/math] is a net in [math]\displaystyle{ X }[/math] that order-converges to [math]\displaystyle{ x_0 }[/math] in [math]\displaystyle{ X, }[/math] then the net [math]\displaystyle{ \left(T\left(x_{\alpha}\right)\right)_{\alpha \in A} }[/math] order-converges to [math]\displaystyle{ T\left(x_0\right) }[/math] in [math]\displaystyle{ Y. }[/math] [math]\displaystyle{ T }[/math] is said to be sequentially order continuous if whenever [math]\displaystyle{ \left(x_n\right)_{n \in \N} }[/math] is a sequence in [math]\displaystyle{ X }[/math] that order-converges to [math]\displaystyle{ x_0 }[/math] in [math]\displaystyle{ X, }[/math]then the sequence [math]\displaystyle{ \left(T\left(x_n\right)\right)_{n \in \N} }[/math] order-converges to [math]\displaystyle{ T\left(x_0\right) }[/math] in [math]\displaystyle{ Y. }[/math]

Related results

In an order complete vector lattice [math]\displaystyle{ X }[/math] whose order is regular, [math]\displaystyle{ X }[/math] is of minimal type if and only if every order convergent filter in [math]\displaystyle{ X }[/math] converges when [math]\displaystyle{ X }[/math] is endowed with the order topology.

See also

• Banach lattice – Banach space with a compatible structure of a lattice
• Fréchet lattice

References

• Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. 
• 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.