Английская Википедия:Abstract m-space

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Шаблон:Short descriptionIn mathematics, specifically in order theory and functional analysis, an abstract m-space or an AM-space is a Banach lattice <math>(X, \| \cdot \|)</math> whose norm satisfies <math>\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 Шаблон:Math in X such that the interval Шаблон:Math is equal to the unit ball of X; such an element u is unique and an order unit of X.Шаблон:Sfn

Examples

The strong dual of an AL-space is an AM-space with unit.Шаблон:Sfn

If X is an Archimedean ordered vector lattice, u is an order unit of X, and p_u is the Minkowski functional of <math>[u, -u] := \{ x \in X : -u \leq x \text{ and } x \leq x \},</math> then the complete of the semi-normed space (X, p_u) is an AM-space with unit u.Шаблон:Sfn

Properties

Every AM-space is isomorphic (as a Banach lattice) with some closed vector sublattice of some suitable <math>C_{\R}\left( X \right)</math>.Шаблон:Sfn The strong dual of an AM-space with unit is an AL-space.Шаблон:Sfn

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>\sigma\left( X^{\prime}, X \right)</math>-compact) subset of <math>X^{\prime}</math> and furthermore, the evaluation map <math>I : X \to C_{\R} \left( K \right)</math> defined by <math>I(x) := I_x</math> (where <math>I_x : K \to \R</math> is defined by <math>I_x(t) = \langle x, t \rangle</math>) is an isomorphism.Шаблон:Sfn

See also

• Vector lattice
• AL-space

Bibliography

• Шаблон:Schaefer Wolff Topological Vector Spaces