Abstract m-space
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 in X such that the interval is equal to the unit ball of X;
such an element u is unique and an order unit of X.
Examples
The strong dual of an AL-space is an AM-space with unit.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 is an AM-space with unit u.
Properties
Every AM-space is isomorphic with some closed vector sublattice of some suitable.The strong dual of an AM-space with unit is an AL-space.
If X ≠ 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 subset of and furthermore, the evaluation map defined by is an isomorphism.