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] := { 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]


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. -compact) subset of and furthermore, the evaluation map defined by (where is defined by ) is an isomorphism.[1]

See also


  1. ^ a b c d e f Schaefer & Wolff 1999, pp. 242–250.