Riesz space

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.