In mathematics, specifically in order theory and functional analysis, an abstract L-space, an AL-space, or an abstract Lebesgue space is a Banach lattice
[2] The strong dual of an AM-space with unit is an AL-space.
[1] The reason for the name abstract L-space is because every AL-space is isomorphic (as a Banach lattice) with some subspace of
[1] Every AL-space X is an order complete vector lattice of minimal type; however, the order dual of X, denoted by X+, is not of minimal type unless X is finite-dimensional.
[1] The strong dual of an AL-space is an AM-space with unit.
, where K is a compact extremally disconnected topological space; furthermore, under the evaluation map, X is isomorphic with the band of all real Radon measures 𝜇 on K such that for every majorized and directed subset S of