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 , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} 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 ( μ ) . {\displaystyle 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 ′ {\displaystyle X^{\prime }} (which is equal to X+) of an AL-space X is a Banach lattice that can be identified with C R ( K ) {\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 C R ( K ) , {\displaystyle C_{\mathbb {R} }(K),} we have lim f ∈ S μ ( f ) = μ ( sup S ) . {\displaystyle \lim _{f\in S}\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
  • 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 .