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 (that is, a subset contained in an interval of the form

is the set of all order bounded subsets of X, in which case this common value is called the order limit of

[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.

α

α ∈

in a vector lattice

α ≤ β

α

α

: α ∈

α

α ∈

in a vector lattice

α

α ∈

α

α

α ∈

[2] A linear map

between vector lattices is said to be order continuous if whenever

α

α ∈

α

is said to be sequentially order continuous if whenever

[2] In an order complete vector lattice

whose order is regular,

is of minimal type if and only if every order convergent filter in

is endowed with the order topology.