Abstract L-space

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 $(X,\Vert \cdot \Vert )$ whose norm is additive on the positive cone of X.

In probability theory, it means the standard probability space.

Examples

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

Properties

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

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

See also

• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets
• AM-space – Concept in order theoryPages displaying short descriptions of redirect targets