Order convergence

In mathematics, specifically in order theory and functional analysis, a filter ${\displaystyle {\mathcal {F}}}$ in an order complete vector lattice ${\displaystyle X}$ is order convergent if it contains an order bounded subset (that is, a subset contained in an interval of the form ${\displaystyle [a,b]:=\{x\in X:a\leq x{\text{ and }}x\leq b\}}$) and if $${\displaystyle \sup \left\{\inf S:S\in \operatorname {OBound} (X)\cap {\mathcal {F}}\right\}=\inf \left\{\sup S:S\in \operatorname {OBound} (X)\cap {\mathcal {F}}\right\},}$$ where ${\displaystyle \operatorname {OBound} (X)}$ is the set of all order bounded subsets of X, in which case this common value is called the order limit of ${\displaystyle {\mathcal {F}}}$ in ${\displaystyle X.}$ ^[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 ${\displaystyle X}$ 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 ${\displaystyle X.}$ A net ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ in a vector lattice ${\displaystyle X}$ 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 ${\displaystyle X}$ that decreases to ${\displaystyle 0}$ and satisfies ${\displaystyle \left|x_{\alpha }-x_{0}\right|\leq y_{\alpha }}$ for all ${\displaystyle \alpha \in A}$. ^[2]

Order continuity

A linear map ${\displaystyle T:X\to Y}$ between vector lattices is said to be order continuous if whenever ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ is a net in ${\displaystyle X}$ that order-converges to ${\displaystyle x_{0}}$ in ${\displaystyle X,}$ then the net ${\displaystyle \left(T\left(x_{\alpha }\right)\right)_{\alpha \in A}}$ order-converges to ${\displaystyle T\left(x_{0}\right)}$ in ${\displaystyle Y.}$${\displaystyle T}$ is said to be sequentially order continuous if whenever ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ is a sequence in ${\displaystyle X}$ that order-converges to ${\displaystyle x_{0}}$ in ${\displaystyle X,}$then the sequence ${\displaystyle \left(T\left(x_{n}\right)\right)_{n\in \mathbb {N} }}$ order-converges to ${\displaystyle T\left(x_{0}\right)}$ in ${\displaystyle Y.}$ ^[2]

Related results

In an order complete vector lattice ${\displaystyle X}$ whose order is regular, ${\displaystyle X}$ is of minimal type if and only if every order convergent filter in ${\displaystyle X}$ converges when ${\displaystyle X}$ is endowed with the order topology. ^[1]

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

1. Schaefer & Wolff 1999, pp. 234–242.
2. Khaleelulla 1982, p. 8.

• 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.