Pistia

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]

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 p_u is the Minkowski functional of then the complete of the semi-normed space (X, p_u) 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 .^[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

• Vector lattice
• AL-space

References

Bibliography

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.