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 ${\displaystyle (X,\|\cdot \|)}$ whose norm is additive on the positive cone of X.[1]

In probability theory, it means the standard probability space.[2]

## Examples

The strong dual of an AM-space with unit is an AL-space.[1]

## Properties

The reason for the name abstract L-space is because every AL-space is isomorphic (as a Banach lattice) with some subspace of ${\displaystyle L^{1}(\mu ).}$[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] Each order interval in an AL-space is weakly compact.[1]

The strong dual of an AL-space is an AM-space with unit.[1] The continuous dual space ${\displaystyle X^{\prime ))$ (which is equal to X+) of an AL-space X is a Banach lattice that can be identified with ${\displaystyle C_{\mathbb {R} }(K)}$, 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 ${\displaystyle C_{\mathbb {R} }(K),}$ we have ${\displaystyle \lim _{f\in S}\mu (f)=\mu (\sup S).}$[1]

• Vector lattice – Partially ordered vector space, ordered as a lattice
• AM-space – Concept in order theory

## References

1. Schaefer & Wolff 1999, pp. 242–250.
2. ^ Takeyuki Hida, Stationary Stochastic Processes, p. 21