Order convergence

In mathematics, specifically in order theory and functional analysis, a filter ${\displaystyle {\mathcal {F}}}$ in an order complete vector lattice X is order convergent if it contains an order bounded subset (i.e. is contained in an interval of the form [a,b] = { x ∈ X : a ≤ x ≤ b }) and if ${\displaystyle {\mathcal {F}}}$,

${\displaystyle \sup \left\{\inf S:S\in \operatorname {OBound} \left(X\right)\cap {\mathcal {F}}\right\}=\inf \left\{\sup S:S\in \operatorname {OBound} \left(X\right)\cap {\mathcal {F}}\right\}}$,

where ${\displaystyle \operatorname {OBound} \left(X\right)}$ is the set of all order bounded subsets of X, in which case this common value is called the order limit of ${\displaystyle {\mathcal {F}}}$ (in X).[1]

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 ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ in a vector lattice X is said to decrease to ${\displaystyle x_{0}\in X}$ if ${\displaystyle \alpha \leq \beta }$ implies ${\displaystyle x_{\beta }\leq x_{\alpha }}$ and ${\displaystyle x_{0}=inf\left\{x_{\alpha }:\alpha \in A\right\}}$ in X. A net ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ in a vector lattice X is said to order-converge to ${\displaystyle x_{0}\in X}$ if there is a net ${\displaystyle \left(y_{\alpha }\right)_{\alpha \in A}}$ in X that decreases to 0 and satisfies ${\displaystyle \left|x_{\alpha }-x_{0}\right|\leq y_{\alpha }}$ for all ${\displaystyle \alpha \in A}$.[2]

Order continuity

A linear T : X → Y between vector lattices is said to be order continuous if whenever ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ is a net in X that order-converges to x₀ in X, then the net ${\displaystyle \left(T\left(x_{\alpha }\right)\right)_{\alpha \in A}}$ order-converges to T(x₀) in Y. T is said to be sequentially order continuous if whenever ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ is a sequence in X that order-converges to x₀ in X, then the sequence ${\displaystyle \left(T\left(x_{n}\right)\right)_{n\in \mathbb {N} }}$ order-converges to T(x₀) in Y.[2]

Related results

In n order complete vector lattice X whose order is regular, X is of minimal type if and only if every order convergent filter in X converges when X is endowed with the order topology.[1]

See also

• Banach lattice
• Fréchet lattice
• Locally convex lattice
• Normed lattice
• Vector lattice

References

1. Schaefer & Wolff 1999, pp. 234–242. sfn error: no target: CITEREFSchaeferWolff1999 (help)
2. Khaleelulla 1982, p. 8. sfn error: no target: CITEREFKhaleelulla1982 (help)

• 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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• 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.CS1 maint: ref=harv (link)