In mathematics, specifically in order theory and functional analysis, a filter F {\displaystyle {\mathcal {F}}} in an order complete vector lattice X {\displaystyle 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 ∈ X : a ≤ x and x ≤ b } {\displaystyle [a,b]:=\{x\in X:a\leq x{\text{ and }}x\leq b\}} ) and if
sup { inf S : S ∈ OBound ( X ) ∩ F } = inf { sup S : S ∈ OBound ( X ) ∩ F } , {\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 OBound ( X ) {\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 F {\displaystyle {\mathcal {F}}} in X . {\displaystyle 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 ( x α ) α ∈ A {\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}} in a vector lattice X {\displaystyle X} is said to decrease to x 0 ∈ X {\displaystyle x_{0}\in X} if α ≤ β {\displaystyle \alpha \leq \beta } implies x β ≤ x α {\displaystyle x_{\beta }\leq x_{\alpha }} and x 0 = i n f { x α : α ∈ A } {\displaystyle x_{0}=inf\left\{x_{\alpha }:\alpha \in A\right\}} in X . {\displaystyle X.} A net ( x α ) α ∈ A {\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}} in a vector lattice X {\displaystyle X} is said to order-converge to x 0 ∈ X {\displaystyle x_{0}\in X} if there is a net ( y α ) α ∈ A {\displaystyle \left(y_{\alpha }\right)_{\alpha \in A}} in X {\displaystyle X} that decreases to 0 {\displaystyle 0} and satisfies | x α − x 0 | ≤ y α {\displaystyle \left|x_{\alpha }-x_{0}\right|\leq y_{\alpha }} for all α ∈ A {\displaystyle \alpha \in A} .
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