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]

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 ${\displaystyle C_{\mathbb {R} }\left(K\right)}$, 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 ${\displaystyle C_{\mathbb {R} }\left(K\right)}$, we have ${\displaystyle \lim _{f\in S}\mu \left(f\right)=\mu \left(\sup S\right)}$.[1]

See also

• Vector lattice
• AM-space

References

1. Schaefer & Wolff 1999, pp. 242–250.

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)