Abstract L-space explained
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
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
(which is equal to
X+) of an AL-space
X is a
Banach lattice that can be identified with
, 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 on
K such that for every majorized and directed subset
S of
we have
