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 [math]\displaystyle{ (X, \| \cdot \|) }[/math] 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 [math]\displaystyle{ L^1(\mu). }[/math] 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 [math]\displaystyle{ X^{\prime} }[/math] (which is equal to X⁺) of an AL-space X is a Banach lattice that can be identified with [math]\displaystyle{ C_{\R} ( K ) }[/math], 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 [math]\displaystyle{ C_{\R} ( K ), }[/math] we have [math]\displaystyle{ \lim_{f \in S} \mu ( f ) = \mu ( \sup S ). }[/math]

References

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