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.
Definition
Preliminaries If X {\displaystyle X} is an ordered vector space (which by definition is a vector space over the reals) and if S {\displaystyle S} is a subset of X {\displaystyle X} then an element b ∈ X {\displaystyle b\in X} is an upper bound (resp. lower bound) of S {\displaystyle S} if s ≤ b {\displaystyle s\leq b} (resp. s ≥ b {\displaystyle s\geq b} ) for all s ∈ S . {\displaystyle s\in S.}
An element a {\displaystyle a} in X {\displaystyle X} is the least upper bound or supremum (resp. greater lower bound or infimum) of S {\displaystyle S} if it is an upper bound (resp. a lower bound) of S {\displaystyle S} and if for any upper bound (resp. any lower bound) b {\displaystyle b} of S , {\displaystyle S,} a ≤ b {\displaystyle a\leq b} (resp. a ≥ b {\displaystyle a\geq b} ).
Definitions
Preordered vector lattice A preordered vector lattice is a preordered vector space E {\displaystyle E} in which every pair of elements has a supremum. More explicitly, a preordered vector lattice is vector space endowed with a preorder, ≤ , {\displaystyle \,\leq ,\,} such that for any x , y , z ∈ E {\displaystyle x,y,z\in E} :
Translation Invariance: x ≤ y {\displaystyle x\leq y} implies x + z ≤ y + z . {\displaystyle x+z\leq y+z.}
Positive Homogeneity: For any scalar 0 ≤ a , {\displaystyle 0\leq a,} x ≤ y {\displaystyle x\leq y} implies a x ≤ a y . {\displaystyle ax\leq ay.}
For any pair of vectors x , y ∈ E , {\displaystyle x,y\in E,} there exists a supremum (denoted x ∨ y {\displaystyle x\vee y} ) in E {\displaystyle E} with respect to the order ( ≤ ) . {\displaystyle \,(\leq ).\,}
The preorder, together with items 1 and 2, which make it "compatible with the vector space structure", make E {\displaystyle E} 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 E {\displaystyle E} also a meet semilattice, hence a lattice. A preordered vector space E {\displaystyle E} is a preordered vector lattice if and only if it satisfies any of the following equivalent properties:
For any x , y ∈ E , {\displaystyle x,y\in E,} their supremum exists in E . {\displaystyle E.}
For any x , y ∈ E , {\displaystyle x,y\in E,} their infimum exists in E . {\displaystyle E.}
For any x , y ∈ E , {\displaystyle x,y\in E,} their infimum and their supremum exist in E . {\displaystyle E.}
For any x ∈ E , {\displaystyle x\in E,} sup { x , 0 } {\displaystyle \sup\{x,0\}} exists in E . {\displaystyle E.}
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