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 Sur la décomposition des opérations fonctionelles linéaires.
Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis.
If is an ordered vector space (which by definition is a vector space over the reals) and if is a subset of then an element is an upper bound (resp. lower bound) of if (resp. ) for all
An element in is the least upper bound or supremum (resp. greater lower bound or infimum) of if it is an upper bound (resp. a lower bound) of and if for any upper bound (resp. any lower bound) of (resp. ).
A preordered vector lattice is a preordered vector space in which every pair of elements has a supremum.
More explicitly, a preordered vector lattice is vector space endowed with a preorder, such that for any :
Translation Invariance: implies
Positive Homogeneity: For any scalar implies
For any pair of vectors there exists a supremum (denoted ) in with respect to the order
The preorder, together with items 1 and 2, which make it "compatible with the vector space structure", make a preordered vector space.
Item 3 says that the preorder is a join semilattice.
Because the preorder is compatible with the vector space structure, one can show that any pair also have an infimum, making also a meet semilattice, hence a lattice.
A preordered vector space is a preordered vector lattice if and only if it satisfies any of the following equivalent properties:
For any their supremum exists in
For any their infimum exists in
For any their infimum and their supremum exist in
For any exists in
A Riesz space or a vector lattice is a preordered vector lattice whose preorder is a partial order.
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En mathématiques, un espace vectoriel ordonné (ou espace vectoriel partiellement ordonné) est un espace vectoriel sur muni d'une relation d'ordre compatible avec sa structure. Il est dit totalement ordonné si l'ordre associé est un ordre total. Soit E un espace vectoriel sur le corps des réels et un préordre sur .
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 Sur la décomposition des opérations fonctionelles linéaires. Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem.
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