Riesz space
Partially ordered vector space, ordered as a lattice / From Wikipedia, the free encyclopedia
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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.
Preliminaries
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. ).
Definitions
Preordered vector lattice
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:
Riesz space and vector lattices
A Riesz space or a vector lattice is a preordered vector lattice whose preorder is a partial order. Equivalently, it is an ordered vector space for which the ordering is a lattice.
Note that many authors required that a vector lattice be a partially ordered vector space (rather than merely a preordered vector space) while others only require that it be a preordered vector space. We will henceforth assume that every Riesz space and every vector lattice is an ordered vector space but that a preordered vector lattice is not necessarily partially ordered.
If is an ordered vector space over whose positive cone (the elements ) is generating (that is, such that ), and if for every either or exists, then is a vector lattice.[2]
Intervals
An order interval in a partially ordered vector space is a convex set of the form In an ordered real vector space, every interval of the form is balanced.[3] From axioms 1 and 2 above it follows that and implies A subset is said to be order bounded if it is contained in some order interval.[3] An order unit of a preordered vector space is any element such that the set is absorbing.[3]
The set of all linear functionals on a preordered vector space that map every order interval into a bounded set is called the order bound dual of and denoted by [3] If a space is ordered then its order bound dual is a vector subspace of its algebraic dual.
A subset of a vector lattice is called order complete if for every non-empty subset such that is order bounded in both and exist and are elements of We say that a vector lattice is order complete if is an order complete subset of [4]
Finite-dimensional Riesz spaces are entirely classified by the Archimedean property:
- Theorem:[5] Suppose that is a vector lattice of finite-dimension If is Archimedean ordered then it is (a vector lattice) isomorphic to under its canonical order. Otherwise, there exists an integer satisfying such that is isomorphic to where has its canonical order, is with the lexicographical order, and the product of these two spaces has the canonical product order.
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.[6] On the other hand, epi-mono factorization in the category of ℝ-vector spaces also applies to Riesz spaces: every lattice-ordered vector space injects into a quotient of ℝκ by a solid subspace.[7]