Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires.
Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis.
Because the preorder is compatible with the vector space structure, one can show that any pair also have an infimum, making
In an ordered real vector space, every interval of the form
[3] An order unit of a preordered vector space is any element
[3] The set of all linear functionals on a preordered vector space
is called order complete if for every non-empty subset
[4] Finite-dimensional Riesz spaces are entirely classified by the Archimedean property: The same result does not hold in infinite dimensions.
For an example due to Kaplansky, consider the vector space V of functions on [0,1] that are continuous except at finitely many points, where they have a pole of second order.
This space is lattice-ordered by the usual pointwise comparison, but cannot be written as ℝκ for any cardinal κ.
[2] Disjoint complements are always bands, but the converse is not true in general.
[2] Every Riesz space is a distributive lattice; that is, it has the following equivalent[Note 1] properties:[8] for all
There are a number of meaningful non-equivalent ways to define convergence of sequences or nets with respect to the order structure of a Riesz space.
is defined to be an ideal with the extra property, that for any element
is the supremum of an arbitrary subset of positive elements in
in a Riesz space, is called a projection band, if
There then also exists a positive linear idempotent, or projection,
The collection of all projection bands in a Riesz space forms a Boolean algebra.
A vector lattice is complete if every subset has both a supremum and an infimum.
A vector lattice is Dedekind complete if each set with an upper bound has a supremum and each set with a lower bound has an infimum.
is the preorder induced by the pointed convex cone
that induces a canonical preordering on the quotient space
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
are two non-trivial ordered vector spaces with respective positive cones
The set of all positive linear forms on a vector space, denoted by
there do exist ordered vector spaces for which set equality does not hold.
is linear and if any one of the following equivalent conditions hold:[9][5] A pre-ordered vector lattice homomorphism that is bijective is a pre-ordered vector lattice isomorphism.
is a non-zero linear functional on a vector lattice
[5] There are numerous projection properties that Riesz spaces may have.