Loading AI tools
Concept in order theory From Wikipedia, the free encyclopedia
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. -compact) subset of and furthermore, the evaluation map defined by (where is defined by ) is an isomorphism.[1]
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.