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 $(X, \, \cdot \, )$ 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 $L^1(\mu).$
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 $X^$ (which is equal to ''X''⁺) of an AL-space ''X'' is a Banach lattice that can be identified with $C_ ( K )$, 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 $C_ ( K ),$ we have $\lim_ \mu ( f ) = \mu ( \sup S ).$

See also

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References

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{{Ordered topological vector spaces

Functional analysis