Order convergence

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

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

Order continuity

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

Related results

In an 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.

See also

• Banach lattice – Banach space with a compatible structure of a lattice
• Fréchet lattice – Topological vector lattice
• Locally convex 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.