Order convergence explained

l{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 and x \leq b \} ) and if $$\backslash sup\; \backslash left\backslash \; =\; \backslash inf\; \backslash left\backslash ,$$ 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 l{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 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 vector lattice X is said to order-converge to x _{0} \in X if there is a net \left(y _{\alpha} \right) _{\alpha} 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} is a net in X that order-converges to x _{0} in X, then the net \left(T\left(x _{\alpha} \right)\right) _{\alpha} 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} is a sequence in X that order-converges to x _{0} in X, then the sequence \left(T\left(x _{n\right)\right)} _{n} order-converges to T\left(x _{0\right)} in Y.

Related results

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