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.

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 $X^{\prime }$ (which is equal to X⁺) of an AL-space X is a Banach lattice that can be identified with $C_{\mathbb {R} }\left(K\right)$, 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 $C_{\mathbb {R} }\left(K\right)$, we have $\lim _{f\in S}\mu \left(f\right)=\mu \left(\sup S\right)$.

See also

• Vector lattice
• AM-space

References

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)