In mathematics, specifically in

order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

and

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, an abstract ''m''-space or an AM-space is a

Banach lattice In the mathematical disciplines of in functional analysis and order theory, a Banach lattice is a complete normed vector space with a lattice order, \leq, such that for all , the implication \Rightarrow holds, where the absolute value is defin ...

(X, \,  \cdot \, )

whose norm satisfies

\left\,  \sup \ \right\,  = \sup \left\

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 in ''X'' such that the interval is equal to the unit ball of ''X''; such an element ''u'' is unique and an

order unit An order unit is an element of an ordered vector space which can be used to bound all elements from above. In this way (as seen in the first example below) the order unit generalizes the unit element in the reals. According to H. H. Schaefer, "mo ...

of ''X''.

Examples

The strong dual of an AL-space is an AM-space with unit. If ''X'' is an

Archimedean order In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, as ty ...

vector lattice In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Su ...

, ''u'' is an

of ''X'', and ''p''_''u'' is the

Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, ...

, -u := \,

then the complete of the semi-normed space (''X'', ''p''_''u'') is an AM-space with unit ''u''.

Properties

Every AM-space is isomorphic (as a Banach lattice) with some closed vector sublattice of some suitable

C_\left( X \right)

. The strong dual of an AM-space with unit is an AL-space. If ''X'' ≠ 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.

\sigma\left( X^, X \right)

-compact) subset of

X^

and furthermore, the evaluation map

I : X \to C_ \left( K \right)

defined by

I(x) := I_x

(where

I_x : K \to \R

is defined by

I_x(t) = \langle x, t \rangle

) is an isomorphism.

References

Bibliography

* {{Ordered topological vector spaces Functional analysis

Examples

Properties

See also

References

Bibliography