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 ${\displaystyle (X,\|\cdot \|)}$ 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]

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

References

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.