Order convergence


In mathematics, specifically in order theory and functional analysis, a filter in an order complete vector lattice X is order convergent if it contains an order bounded subset and if,
where is the set of all order bounded subsets of X, in which case this common value is called the order limit of .
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 in a vector lattice X is said to decrease to if implies and in X.
A net in a vector lattice X is said to order-converge to if there is a net in X that decreases to 0 and satisfies for all.

Order continuity

A linear T : X → Y between vector lattices is said to be order continuous if whenever is a net in X that order-converges to x0 in X, then the net order-converges to T in Y.
T is said to be sequentially order continuous if whenever is a sequence in X that order-converges to x0 in X, then the sequence order-converges to T in Y.

Related results

In n 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.