Ordered vector space

is a partial order compatible with the vector space structure of

The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping

A cone is called pointed if it contains the origin.

The intersection of any non-empty family of cones (resp.

the positive cone of this resulting preordered vector space is

There is thus a one-to-one correspondence between pointed convex cones and vector preorders on

and furthermore, the positive cone of this ordered vector space will be

Therefore, there exists a one-to-one correspondence between the proper convex cones of

is in one-to-one correspondence with the family of all proper cones that are maximal under set inclusion.

are two orderings of a vector space with positive cones

considered as a vector space over the reals with the lexicographic ordering forms a preordered vector space whose order is Archimedean if and only if

is a vector space (over the reals) of real-valued functions on

of all measurable almost-everywhere bounded real-valued maps on

[3] An order interval in a preordered vector space is set of the form

[2] An order unit of a preordered vector space is any element

[2] The set of all linear functionals on a preordered vector space

is called order complete if for every non-empty subset

is a preordered vector space over the reals with order unit

are two non-trivial ordered vector spaces with respective positive cones

on a preordered vector space is called positive if it satisfies either of the following equivalent conditions: The set of all positive linear forms on a vector space with positive cone

The preorder induced by the dual cone on the space of linear functionals on

there do exist ordered vector spaces for which set equality does not hold.

[2] A topological vector space (TVS) that is an ordered vector space is necessarily Archimedean if its positive cone is closed.

[2] This property guarantees that there are sufficiently many positive linear forms to be able to successfully use the tools of duality to study ordered vector spaces.

be a preordered vector space with positive cone

is the partial order induced by the pointed convex cone

that induces a canonical preordering on the quotient space

is also a topological vector space (TVS) and if for each neighborhood

is a family of preordered vector spaces and that the positive cone of

is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of