In mathematics, specifically in order theory and functional analysis, an abstract m-space or an AM-space is a Banach lattice
whose norm satisfies
for all x and y in the positive cone of X.
We say that an AM-space X is an AM-space with unit if in addition there exists some u ≥ 0 in X such that the interval [−u, u] := { z ∈ X : −u ≤ z and z ≤ u } is equal to the unit ball of X; such an element u is unique and an order unit of X.
[1] The strong dual of an AL-space is an AM-space with unit.
[1] If X is an Archimedean ordered vector lattice, u is an order unit of X, and pu is the Minkowski functional of
then the complete of the semi-normed space (X, pu) is an AM-space with unit u.
[1] Every AM-space is isomorphic (as a Banach lattice) with some closed vector sublattice of some suitable
[1] The strong dual of an AM-space with unit is an AL-space.
[1] If X ≠ { 0 } is an AM-space with unit then the set K of all extreme points of the positive face of the dual unit ball is a non-empty and weakly compact (i.e.
σ
and furthermore, the evaluation map