Order convergence

In mathematics, specifically in order theory and functional analysis, a filter ${\mathcal {F}}$ 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 $[a,b]:=\{x\in X:a\leq x{\text{ and }}x\leq b\}$) and if

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

where $\operatorname {OBound} (X)$ is the set of all order bounded subsets of X, in which case this common value is called the order limit of ${\mathcal {F}}$ 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 $\alpha \leq \beta$ implies $x_{\beta }\leq x_{\alpha }$ and $x_{0}=inf\left\{x_{\alpha }:\alpha \in A\right\}$ 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 $\left|x_{\alpha }-x_{0}\right|\leq y_{\alpha }$ 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 $\left(x_{n}\right)_{n\in \mathbb {N} }$ is a sequence in $X$ that order-converges to $x_{0}$ in $X,$then the sequence $\left(T\left(x_{n}\right)\right)_{n\in \mathbb {N} }$ 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
• Normed lattice
• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

References

• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.