Riesz space

Finite-dimensional Riesz spaces are entirely classified by the Archimedean property:

Theorem:[5] Suppose that ${\displaystyle X}$ is a vector lattice of finite-dimension ${\displaystyle n.}$ If ${\displaystyle X}$ is Archimedean ordered then it is (a vector lattice) isomorphic to ${\displaystyle \mathbb {R} ^{n}}$ under its canonical order. Otherwise, there exists an integer ${\displaystyle k}$ satisfying ${\displaystyle 2\leq k\leq n}$ such that ${\displaystyle X}$ is isomorphic to ${\displaystyle \mathbb {R} _{L}^{k}\times \mathbb {R} ^{n-k}}$ where ${\displaystyle \mathbb {R} ^{n-k}}$ has its canonical order, ${\displaystyle \mathbb {R} _{L}^{k}}$ is ${\displaystyle \mathbb {R} ^{k}}$ 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 ℝ^κ 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 ℝ^κ by a solid subspace.[7]

Every Riesz space is a partially ordered vector space, but not every partially ordered vector space is a Riesz space.

Note that for any subset ${\displaystyle A}$ of ${\displaystyle X,}$ ${\displaystyle \sup A=-\inf(-A)}$ whenever either the supremum or infimum exists (in which case they both exist).[2] If ${\displaystyle x\geq 0}$ and ${\displaystyle y\geq 0}$ then ${\displaystyle [0,x]+[0,y]=[0,x+y].}$[2] For all ${\displaystyle a,b,x,{\text{ and }}y}$ in a Riesz space ${\displaystyle X,}$ ${\displaystyle a-\inf(x,y)+b=\sup(a-x+b,a-y+b).}$[4]

There are a number of meaningful non-equivalent ways to define convergence of sequences or nets with respect to the order structure of a Riesz space. A sequence ${\displaystyle \left\{x_{n}\right\}}$ in a Riesz space ${\displaystyle E}$ is said to converge monotonely if it is a monotone decreasing (resp. increasing) sequence and its infimum (supremum) ${\displaystyle x}$ exists in ${\displaystyle E}$ and denoted ${\displaystyle x_{n}\downarrow x}$ (resp. ${\displaystyle x_{n}\uparrow x}$).

A sequence ${\displaystyle \left\{x_{n}\right\}}$ in a Riesz space ${\displaystyle E}$ is said to converge in order to ${\displaystyle x}$ if there exists a monotone converging sequence ${\displaystyle \left\{p_{n}\right\}}$ in ${\displaystyle E}$ such that ${\displaystyle \left|x_{n}-x\right|<p_{n}\downarrow 0.}$

If ${\displaystyle u}$ is a positive element of a Riesz space ${\displaystyle E}$ then a sequence ${\displaystyle \left\{x_{n}\right\}}$ in ${\displaystyle E}$ is said to converge u-uniformly to ${\displaystyle x}$ if for any ${\displaystyle r>0}$ there exists an ${\displaystyle N}$ such that ${\displaystyle \left|x_{n}-x\right|<ru}$ for all ${\displaystyle n>N.}$

The extra structure provided by these spaces provide for distinct kinds of Riesz subspaces. The collection of each kind structure in a Riesz space (for example, the collection of all ideals) forms a distributive lattice.

Sublattices

If ${\displaystyle M}$ is a vector subspace of a preordered vector space ${\displaystyle X}$ then the canonical ordering on ${\displaystyle M}$ induced by ${\displaystyle X}$'s positive cone ${\displaystyle C}$ is the preorder induced by the pointed convex cone ${\displaystyle C\cap M,}$ where this cone is proper if ${\displaystyle C}$ is proper (that is, if ${\displaystyle C\cap (-C)=\varnothing }$).[3]

A sublattice of a vector lattice ${\displaystyle X}$ is a vector subspace ${\displaystyle M}$ of ${\displaystyle X}$ such that for all ${\displaystyle x,y\in M,}$ ${\displaystyle \sup _{}{}_{X}(x,y)}$ belongs to ${\displaystyle X}$ (importantly, note that this supremum is taken in ${\displaystyle X}$ and not in ${\displaystyle M}$).[3] If ${\displaystyle X=L^{p}([0,1],\mu )}$ with ${\displaystyle 0<p<1,}$ then the 2-dimensional vector subspace ${\displaystyle M}$ of ${\displaystyle X}$ defined by all maps of the form ${\displaystyle t\mapsto at+b}$ (where ${\displaystyle a,b\in \mathbb {R} }$) is a vector lattice under the induced order but is not a sublattice of ${\displaystyle X.}$[5] This despite ${\displaystyle X}$ being an order complete Archimedean ordered topological vector lattice. Furthermore, there exist vector a vector sublattice ${\displaystyle N}$ of this space ${\displaystyle X}$ such that ${\displaystyle N\cap C}$ has empty interior in ${\displaystyle X}$ but no positive linear functional on ${\displaystyle N}$ can be extended to a positive linear functional on ${\displaystyle X.}$[5]

Quotient lattices

Let ${\displaystyle M}$ be a vector subspace of an ordered vector space ${\displaystyle X}$ having positive cone ${\displaystyle C,}$ let ${\displaystyle \pi :X\to X/M}$ be the canonical projection, and let ${\displaystyle {\hat {C}}:=\pi (C).}$ Then ${\displaystyle {\hat {C}}}$ is a cone in ${\displaystyle X/M}$ that induces a canonical preordering on the quotient space ${\displaystyle X/M.}$ If ${\displaystyle {\hat {C}}}$ is a proper cone in ${\displaystyle X/M}$ then ${\displaystyle {\hat {C}}}$ makes ${\displaystyle X/M}$ into an ordered vector space.[3] If ${\displaystyle M}$ is ${\displaystyle C}$-saturated then ${\displaystyle {\hat {C}}}$ defines the canonical order of ${\displaystyle X/M.}$[5] Note that ${\displaystyle X=\mathbb {R} _{0}^{2}}$ provides an example of an ordered vector space where ${\displaystyle \pi (C)}$ is not a proper cone.

If ${\displaystyle X}$ is a vector lattice and ${\displaystyle N}$ is a solid vector subspace of ${\displaystyle X}$ then ${\displaystyle {\hat {C}}}$ defines the canonical order of ${\displaystyle X/M}$ under which ${\displaystyle L/M}$ is a vector lattice and the canonical map ${\displaystyle \pi :X\to X/M}$ is a vector lattice homomorphism. Furthermore, if ${\displaystyle X}$ is order complete and ${\displaystyle M}$ is a band in ${\displaystyle X}$ then ${\displaystyle X/M}$ is isomorphic with ${\displaystyle M^{\bot }.}$[5] Also, if ${\displaystyle M}$ is solid then the order topology of ${\displaystyle X/M}$ is the quotient of the order topology on ${\displaystyle X.}$[5]

If ${\displaystyle X}$ is a topological vector lattice and ${\displaystyle M}$ is a closed solid sublattice of ${\displaystyle X}$ then ${\displaystyle X/L}$ is also a topological vector lattice.[5]

Product

If ${\displaystyle S}$ is any set then the space ${\displaystyle X^{S}}$ of all functions from ${\displaystyle S}$ into ${\displaystyle X}$ is canonically ordered by the proper cone ${\displaystyle \left\{f\in X^{S}:f(s)\in C{\text{ for all }}s\in S\right\}.}$[3]

Suppose that ${\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}}$ is a family of preordered vector spaces and that the positive cone of ${\displaystyle X_{\alpha }}$ is ${\displaystyle C_{\alpha }.}$ Then ${\displaystyle C:=\prod _{\alpha }C_{\alpha }}$ is a pointed convex cone in ${\displaystyle \prod _{\alpha }X_{\alpha },}$ which determines a canonical ordering on ${\displaystyle \prod _{\alpha }X_{\alpha }}$; ${\displaystyle C}$ is a proper cone if all ${\displaystyle C_{\alpha }}$ are proper cones.[3]

Algebraic direct sum

The algebraic direct sum ${\displaystyle \bigoplus _{\alpha }X_{\alpha }}$ of ${\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}}$ is a vector subspace of ${\displaystyle \prod _{\alpha }X_{\alpha }}$ that is given the canonical subspace ordering inherited from ${\displaystyle \prod _{\alpha }X_{\alpha }.}$[3] If ${\displaystyle X_{1},\ldots ,X_{n}}$ are ordered vector subspaces of an ordered vector space ${\displaystyle X}$ then ${\displaystyle X}$ is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of ${\displaystyle X}$ onto ${\displaystyle \prod _{\alpha }X_{\alpha }}$ (with the canonical product order) is an order isomorphism.[3]

A cone ${\displaystyle C}$ in a vector space ${\displaystyle X}$ is said to be generating if ${\displaystyle C-C}$ is equal to the whole vector space.[3] If ${\displaystyle X}$ and ${\displaystyle W}$ are two non-trivial ordered vector spaces with respective positive cones ${\displaystyle P}$ and ${\displaystyle Q,}$ then ${\displaystyle P}$ is generating in ${\displaystyle X}$ if and only if the set ${\displaystyle C=\{u\in \operatorname {L} (X;W):u(P)\subseteq Q\}}$ is a proper cone in ${\displaystyle \operatorname {L} (X;W),}$ which is the space of all linear maps from ${\displaystyle X}$ into ${\displaystyle W.}$ In this case the ordering defined by ${\displaystyle C}$ is called the canonical ordering of ${\displaystyle \operatorname {L} (X;W).}$[3] More generally, if ${\displaystyle M}$ is any vector subspace of ${\displaystyle \operatorname {L} (X;W)}$ such that ${\displaystyle C\cap M}$ is a proper cone, the ordering defined by ${\displaystyle C\cap M}$ is called the canonical ordering of ${\displaystyle M.}$[3]

A linear map ${\displaystyle u:X\to Y}$ between two preordered vector spaces ${\displaystyle X}$ and ${\displaystyle Y}$ with respective positive cones ${\displaystyle C}$ and ${\displaystyle D}$ is called positive if ${\displaystyle u(X)\subseteq D.}$ If ${\displaystyle X}$ and ${\displaystyle Y}$ are vector lattices with ${\displaystyle Y}$ order complete and if ${\displaystyle H}$ is the set of all positive linear maps from ${\displaystyle X}$ into ${\displaystyle Y}$ then the subspace ${\displaystyle M:=H-H}$ of ${\displaystyle \operatorname {L} (X;Y)}$ is an order complete vector lattice under its canonical order; furthermore, ${\displaystyle M}$ contains exactly those linear maps that map order intervals of ${\displaystyle X}$ into order intervals of ${\displaystyle Y.}$[5]

Suppose that ${\displaystyle X}$ and ${\displaystyle Y}$ are preordered vector lattices with positive cones ${\displaystyle C}$ and ${\displaystyle D}$ and let ${\displaystyle u:X\to Y}$ be a map. Then ${\displaystyle u}$ is a preordered vector lattice homomorphism if ${\displaystyle u}$ is linear and if any one of the following equivalent conditions hold:[9][5]

1. ${\displaystyle u}$ preserves the lattice operations
2. ${\displaystyle u(\sup\{x,y\})=\sup\{u(x),u(y)\}}$ for all ${\displaystyle x,y\in X.}$
3. ${\displaystyle u(\inf\{x,y\})=\inf\{u(x),u(y)\}}$ for all ${\displaystyle x,y\in X.}$
4. ${\displaystyle u(|x|)=\sup \left\{u\left(x^{+}\right),u\left(x^{-}\right)\right\}}$ for all ${\displaystyle x\in X.}$
5. ${\displaystyle 0=\inf \left\{u\left(x^{+}\right),u\left(x^{-}\right)\right\}}$ for all ${\displaystyle x\in X.}$
6. ${\displaystyle u(C)=D}$ and ${\displaystyle u^{-1}(0)}$ is a solid subset of ${\displaystyle X.}$[5]
7. if ${\displaystyle x\geq 0}$ then ${\displaystyle u(x)\geq 0.}$[1]
8. ${\displaystyle u}$ is order preserving.[1]

A pre-ordered vector lattice homomorphism that is bijective is a pre-ordered vector lattice isomorphism.

A pre-ordered vector lattice homomorphism between two Riesz spaces is called a vector lattice homomorphism; if it is also bijective, then it is called a vector lattice isomorphism.

If ${\displaystyle u}$ is a non-zero linear functional on a vector lattice ${\displaystyle X}$ with positive cone ${\displaystyle C}$ then the following are equivalent:

1. ${\displaystyle u:X\to \mathbb {R} }$ is a surjective vector lattice homomorphism.
2. ${\displaystyle 0=\inf \left\{u\left(x^{+}\right),u\left(x^{-}\right)\right\}}$ for all ${\displaystyle x\in X.}$
3. ${\displaystyle u\geq 0}$ and ${\displaystyle u^{-1}(0)}$ is a solid hyperplane in ${\displaystyle X.}$
4. ${\displaystyle u'}$ generates an extreme ray of the cone ${\displaystyle X^{*}}$ in ${\displaystyle X^{*}.}$

An extreme ray of the cone ${\displaystyle C}$ is a set ${\displaystyle \{rx:r\geq 0\}}$ where ${\displaystyle x\in C,}$ ${\displaystyle x}$ is non-zero, and if ${\displaystyle y\in C}$ is such that ${\displaystyle x-y\in C}$ then ${\displaystyle y=sx}$ for some ${\displaystyle s}$ such that ${\displaystyle 0\leq s\leq 1.}$[9]

A vector lattice homomorphism from ${\displaystyle X}$ into ${\displaystyle Y}$ is a topological homomorphism when ${\displaystyle X}$ and ${\displaystyle Y}$ are given their respective order topologies.[5]

There are numerous projection properties that Riesz spaces may have. A Riesz space is said to have the (principal) projection property if every (principal) band is a projection band.

The so-called main inclusion theorem relates the following additional properties to the (principal) projection property:[10] A Riesz space is…

• Dedekind Complete (DC) if every nonempty set, bounded above, has a supremum;
• Super Dedekind Complete (SDC) if every nonempty set, bounded above, has a countable subset with identical supremum;
• Dedekind ${\displaystyle \sigma }$-complete if every countable nonempty set, bounded above, has a supremum; and
• Archimedean property if, for every pair of positive elements ${\displaystyle x}$ and ${\displaystyle y}$, whenever the inequality ${\displaystyle nx\leq y}$ holds for all integers ${\displaystyle n}$, ${\displaystyle x=0}$.

Then these properties are related as follows. SDC implies DC; DC implies both Dedekind ${\displaystyle \sigma }$-completeness and the projection property; Both Dedekind ${\displaystyle \sigma }$-completeness and the projection property separately imply the principal projection property; and the principal projection property implies the Archimedean property.

None of the reverse implications hold, but Dedekind ${\displaystyle \sigma }$-completeness and the projection property together imply DC.

• The space of continuous real valued functions with compact support on a topological space ${\displaystyle X}$ with the pointwise partial order defined by ${\displaystyle f\leq g}$ when ${\displaystyle f(x)\leq g(x)}$ for all ${\displaystyle x\in X,}$ is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless ${\displaystyle X}$ satisfies further conditions (for example, being extremally disconnected).
• Any ${\displaystyle L^{p}}$ space with the (almost everywhere) pointwise partial order is a Dedekind complete Riesz space.
• The space ${\displaystyle \mathbb {R} ^{2}}$ with the lexicographical order is a non-Archimedean Riesz space.

• Riesz spaces are lattice ordered groups
• Every Riesz space is a distributive lattice

• Convex cone – Mathematical set closed under positive linear combinations
• Infimum and supremum – Greatest lower bound and least upper bound
• Ordered vector space – Vector space with a partial order
• Partially ordered space – Partially ordered topological space

• Bourbaki, Nicolas; Elements of Mathematics: Integration. Chapters 1–6; ISBN 3-540-41129-1
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Riesz, Frigyes; Sur la décomposition des opérations fonctionelles linéaires, Atti congress. internaz. mathematici (Bologna, 1928), 3, Zanichelli (1930) pp. 143–148
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Sobolev, V. I. (2001) [1994], "Riesz space", Encyclopedia of Mathematics, EMS Press, ISBN 978-1-4020-0609-8
• Zaanen, Adriaan C. (1996), Introduction to Operator Theory in Riesz spaces, Springer, ISBN 3-540-61989-5

• Riesz space at the Encyclopedia of Mathematics