Abstract m-space
In mathematics, specifically in order theory and functional analysis, an abstract m-space or an AM-space is a Banach lattice [math]\displaystyle{ (X, \| \cdot \|) }[/math] whose norm satisfies [math]\displaystyle{ \left\| \sup \{ x, y \} \right\| = \sup \left\{ \| x \|, \| y \| \right\} }[/math] 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]
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 [math]\displaystyle{ [u, -u] := \{ x \in X : -u \leq x \text{ and } x \leq x \}, }[/math] 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 [math]\displaystyle{ C_{\R}\left( X \right) }[/math].[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. [math]\displaystyle{ \sigma\left( X^{\prime}, X \right) }[/math]-compact) subset of [math]\displaystyle{ X^{\prime} }[/math] and furthermore, the evaluation map [math]\displaystyle{ I : X \to C_{\R} \left( K \right) }[/math] defined by [math]\displaystyle{ I(x) := I_x }[/math] (where [math]\displaystyle{ I_x : K \to \R }[/math] is defined by [math]\displaystyle{ I_x(t) = \langle x, t \rangle }[/math]) is an isomorphism.[1]
See also
- Vector lattice
- AL-space
References
Bibliography
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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