Riesz space

If X is an ordered vector space and if S is a subset of X then an element b ∈ X is an upper bound (resp. lower bound) of S if s ≤ b (resp. s ≥ b) for all s ∈ S. An element a in X is the least upper bound or supremum (resp. greater lower bound or infimum) of S if it is an upper bound (resp. a lower bound) of S and if for any upper bounded (resp. any lower bound) b of S, we have a ≤ b (resp. a ≥ b).

A preordered vector lattice is a preordered vector space E 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 x, y, z ∈ E:

1. Translation Invariance: x ≤ y implies x + z ≤ y + z.
2. Positive Homogeneity: For any scalar 0 ≤ α, x ≤ y implies αx ≤ αy.
3. For any pair of vectors x, y in E there exists a supremum (denoted x ∨ y) in E with respect to the order (≤).

The preorder, together with items 1 and 2, which make it "compatible with the vector space structure", make E an 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 also a meet semilattice, hence a lattice.

A preordered vector space E is a preordered vector lattice if and only if it satisfies any of the following equivalent properties:

1. For any x, y ∈ E, their supremum exists in E.
2. For any x, y ∈ E, their infimum exists in E.
3. For any x, y ∈ E, their infimum and their supremum exist in E.
4. For any x ∈ E, sup { x, 0 } exists.[1]

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 E is an ordered vector space over ${\displaystyle \mathbb {R} }$ with positive whose positive cone C is generating (i.e. such that E = C - C), and if for every x, y ∈ C either ${\displaystyle \sup\{x,y\}}$ or ${\displaystyle \inf\{x,y\}}$ exists, then E is a vector lattice.[2]

An order interval in a partially ordered vector space is a convex set of the form [a,b] = { x : a ≤ x ≤ b }. In an ordered real vector space, every interval of the form [−x, x] is balanced.[3] From axioms 1 and 2 above it follows that x,y in [a,b] and λ in (0,1) implies λx + (1 − λ)y in [a,b]. 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 x such that the set [−x, x] is absorbing.[3]

The set of all linear functionals on a preordered vector space V that map every order interval into a bounded set is called the order bound dual of V and denoted by V^b[3] If a space is ordered then its order bound dual is a vector subspace of its algebraic dual.

A subset A of a vector lattice E is called order complete if for every non-empty subset B ⊆ A such that B is order bounded in A, both ${\displaystyle \sup B}$ and ${\displaystyle \inf B}$ exist and are elements of A. We say that a vector lattice E is order complete is E is an order complete subset of E.[4]

For every element x in a Riesz space X, the absolute value of x, denoted by ${\displaystyle |x|}$, is defined to be ${\displaystyle |x|:=\sup \left\{x,-x\right\}}$,[4] where this satisfies -|x| ≤ x ≤ |x| and |x| ≥ 0. For any x and y in X and any real number r, we have ${\displaystyle |rx|=|r||x|}$ and ${\displaystyle |x+y|\leq |x|+|y|}$.[4]

We say that two elements x and y in a vector lattice x are lattice disjoint or disjoint if ${\displaystyle \inf \left\{|x|,|y|\right\}=0}$, in which case we write ${\displaystyle x\perp y}$. Two elements x and y are disjoint if and only if ${\displaystyle \sup\{|x|,|y|\}=|x|+|y|}$. If x and y are disjoint then ${\displaystyle |x+y|=|x|+|y|}$ and ${\displaystyle \left(x+y\right)^{+}=x^{+}+y^{+}}$, where for any element z, ${\displaystyle z^{+}:=\sup \left\{z,0\right\}}$ and ${\displaystyle z^{-}:=\sup \left\{-z,0\right\}}$. We say that two sets A and B are disjoint if a and b are disjoint for all a in A and all b in B, in which case we write ${\displaystyle A\perp B}$.[2] If A is the singleton set ${\displaystyle \{a\}}$ then we will write ${\displaystyle a\perp B}$ in place of ${\displaystyle \{a\}\perp B}$. For any set A, we define the disjoint complement to be the set ${\displaystyle A^{\perp }:=\left\{x\in X:x\perp A\right\}}$.[2] Disjoint complements are always bands, but the converse is not true in general. If A is a subset of X such that ${\displaystyle x=\sup A}$ exists, and if B is a subset lattice in X that is disjoint from A, then B is a lattice disjoint from ${\displaystyle \{x\}}$.[2]

For any x in X, let ${\displaystyle x^{+}:=\sup \left\{x,0\right\}}$ and ${\displaystyle x^{-}:=\sup \left\{-x,0\right\}}$, where note that both of these elements are ${\displaystyle \geq 0}$ and ${\displaystyle x=x^{+}-x^{-}}$ with ${\displaystyle |x|=x^{+}+x^{-}}$. Then ${\displaystyle x^{+}}$ and ${\displaystyle x^{-}}$ are disjoint, and ${\displaystyle x=x^{+}-x^{-}}$ is the unique representation of x as the difference of disjoint elements that are ${\displaystyle \geq 0}$.[2] For all x and y in X, ${\displaystyle \left|x^{+}-y^{+}\right|\leq |x-y|}$ and ${\displaystyle x+y=\sup\{x,y\}+\inf\{x,y\}}$.[2] If y ≥ 0 and x ≤ y then x⁺ ≤ y. Moreover, ${\displaystyle x\leq y}$ if and only if ${\displaystyle x^{+}\leq y^{+}}$ and ${\displaystyle x^{-}\leq x^{-1}}$.[2]

Every Riesz space is a distributive lattice; that is, it has the following equivalent properties: for all x, y, and z in X

1. x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
2. x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)[6][7]
3. (x${\displaystyle \wedge }$y)${\displaystyle \vee }$(y${\displaystyle \wedge }$z)${\displaystyle \vee }$(z${\displaystyle \wedge }$x) = (x${\displaystyle \vee }$y)${\displaystyle \wedge }$(y${\displaystyle \vee }$z)${\displaystyle \wedge }$(z${\displaystyle \vee }$x).
4. x${\displaystyle \wedge }$z = y${\displaystyle \wedge }$z and x${\displaystyle \vee }$z = y${\displaystyle \vee }$z always imply x=y.

Every Riesz space has the Riesz decomposition property.

If X is a vector lattice then a vector sublattice is a vector subspace F of X such that for all x and y in F, ${\displaystyle \sup\{x,y\}}$ belongs to F (where this supremum is taken in X).[4] It can happen that a subspace F of X is a vector lattice under its canonical order but is not a vector sublattice of X.[4]

A vector subspace I of a Riesz space E is called an ideal if it is solid, meaning if for  f  ∈ I and g ∈ E, we have: |g| ≤ | f | implies that g ∈ I.[4] The intersection of an arbitrary collection of ideals is again an ideal, which allows for the definition of a smallest ideal containing some non-empty subset A of E, and is called the ideal generated by A. An Ideal generated by a singleton is called a principal ideal.

A band B in a Riesz space E is defined to be an ideal with the extra property, that for any element f in E for which its absolute value | f | is the supremum of an arbitrary subset of positive elements in B, that f is actually in B. σ-Ideals are defined similarly, with the words 'arbitrary subset' replaced with 'countable subset'. Clearly every band is a σ-ideal, but the converse is not true in general.

The intersection of an arbitrary family of bands is again a band. As with ideals, for every non-empty subset A of E, there exists a smallest band containing that subset, called the band generated by A. A band generated by a singleton is called a principal band.

A band B in a Riesz space, is called a projection band, if E = B ⊕ B^⟂, meaning every element f in E, can be written uniquely as a sum of two elements,  f = u + v, with u in B and v in B^⟂. There then also exists a positive linear idempotent, or projection, P_B : E → E, such that P_B( f ) = u.

The collection of all projection bands in a Riesz space forms a Boolean algebra. Some spaces do not have non-trivial projection bands (e.g. C([0, 1])), so this Boolean algebra may be trivial.

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.

An order complete, regularly ordered vector lattice whose canonical image in its order bidual is order complete is called minimal and is said to be of minimal type.[8]

A linear function f on a preordered vector space is called positive if x ≥ 0 implies f(x) ≥ 0. The set of all positive linear forms on a vector space, denoted by ${\displaystyle C^{*}}$, is a cone equal to the polar of −C. The order dual of an ordered vector space X is the set, denoted by ${\displaystyle X^{+}}$, defined by ${\displaystyle X^{+}:=C^{*}-C^{*}}$. Although ${\displaystyle X^{+}\subseteq X^{b}}$, there do exist ordered vector spaces for which set equality does not hold.[3]