In mathematics, specifically in order theory and functional analysis, an abstract m-space or an AM-space is a Banach lattice ${\displaystyle (X,\|\cdot \|)}$ whose norm satisfies ${\displaystyle \left\|\sup\{x,y\}\right\|=\sup \left\{\|x\|,\|y\|\right\))$ 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] := { zX : −uz and zu } is equal to the unit ball of X; such an element u is unique and an order unit of X.[1]

## Examples

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 ${\displaystyle [u,-u]:=\{x\in X:-u\leq x{\text{ and ))x\leq x\},}$ then the complete of the semi-normed space (X, pu) is an AM-space with unit u.[1]

## Properties

Every AM-space is isomorphic (as a Banach lattice) with some closed vector sublattice of some suitable ${\displaystyle C_{\mathbb {R} }\left(X\right)}$.[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. ${\displaystyle \sigma \left(X^{\prime },X\right)}$-compact) subset of ${\displaystyle X^{\prime ))$ and furthermore, the evaluation map ${\displaystyle I:X\to C_{\mathbb {R} }\left(K\right)}$ defined by ${\displaystyle I(x):=I_{x))$ (where ${\displaystyle I_{x}:K\to \mathbb {R} }$ is defined by ${\displaystyle I_{x}(t)=\langle x,t\rangle }$) is an isomorphism.[1]