Order convergence
In mathematics, specifically in order theory and functional analysis, a filter in an order complete vector lattice 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 in
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 is said to decrease to if implies and inA net in a vector lattice is said to order-converge to if there is a net in that decreases to and satisfies for all.
Order continuity
A linear map between vector lattices is said to be order continuous if whenever is a net in that order-converges to in then the net order-converges to inis said to be sequentially order continuous if whenever is a sequence in that order-converges to in then the sequence order-converges to in