{"history":[],"article":{"id":"1dd2642c-9be4-4158-b7fd-d864ec0df78f","title":"Riesz space","slug":"riesz-space","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **Riesz space**, **[lattice](https://bliptext.com/articles/lattice-order)-ordered vector space** or **vector [lattice](https://bliptext.com/articles/lattice-order)** is a [partially ordered vector space](https://bliptext.com/articles/ordered-vector-space) where the [order structure](https://bliptext.com/articles/order-structure) is a [lattice](https://bliptext.com/articles/lattice-order).\nRiesz spaces are named after [Frigyes Riesz](https://bliptext.com/articles/frigyes-riesz) who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires.\nRiesz spaces have wide-ranging applications.\nThey are important in [measure theory](https://bliptext.com/articles/measure-theory), in that important results are special cases of results for Riesz spaces.\nFor example, the [Radon–Nikodym theorem](https://bliptext.com/articles/radon-nikodym-theorem) follows as a special case of the [Freudenthal spectral theorem](https://bliptext.com/articles/freudenthal-spectral-theorem).\nRiesz spaces have also seen application in [mathematical economics](https://bliptext.com/articles/mathematical-economics) through the work of Greek-American economist and mathematician [Charalambos D. Aliprantis](https://bliptext.com/articles/charalambos-d-aliprantis).\n\n# Definition\n\n## Preliminaries\nIf X is an [ordered vector space](https://bliptext.com/articles/ordered-vector-space) (which by definition is a vector space over the [reals](https://bliptext.com/articles/real-number)) and if S is a subset of X then an element b \\in X is an **upper bound** (resp. **lower bound**) of S if s \\leq b (resp. s \\geq b) for all s \\in S. An element a in X is the least **upper bound** or **[supremum](https://bliptext.com/articles/supremum)** (resp. greater **lower bound** or **[infimum](https://bliptext.com/articles/infimum)**) of S if it is an **upper bound** (resp. a **lower bound**) of S and if for any **upper bound** (resp. any **lower bound**) b of S, a \\leq b (resp. a \\geq b).\n\n## Definitions\n\n## Preordered vector lattice\nA **preordered vector lattice** is a pre[ordered vector space](https://bliptext.com/articles/ordered-vector-space) E in which every pair of elements has a [supremum](https://bliptext.com/articles/supremum).\nMore explicitly, a **[preorder](https://bliptext.com/articles/preorder)ed vector lattice** is vector space endowed with a [preorder](https://bliptext.com/articles/preorder), \\,\\leq,\\, such that for any :\nThe preorder, together with items 1 and 2, which make it \"compatible with the vector space structure\", make E a preordered vector space.\nItem 3 says that the preorder is a [join semilattice](https://bliptext.com/articles/join-semilattice).\nBecause the preorder is compatible with the vector space structure, one can show that any pair also have an [infimum](https://bliptext.com/articles/infimum), making E also a [meet semilattice](https://bliptext.com/articles/meet-semilattice), hence a lattice.\nA preordered vector space E is a preordered vector lattice if and only if it satisfies any of the following equivalent properties:\n

For any x, y \\in E, their [supremum](https://bliptext.com/articles/supremum) exists in E.
For any x, y \\in E, their [infimum](https://bliptext.com/articles/infimum) exists in E.
For any x, y \\in E, their infimum and their supremum exist in E.
For any x \\in E, exists in E.

\n\n## Riesz space and vector lattices\nA **Riesz space** or a **vector lattice** is a preordered **vector lattice** whose preorder is a [partial order](https://bliptext.com/articles/partial-order).\nEquivalently, it is an [ordered vector space](https://bliptext.com/articles/ordered-vector-space)\nfor which the ordering is a [lattice](https://bliptext.com/articles/lattice-order).\nNote that many authors required that a vector lattice be a [partially ordered](https://bliptext.com/articles/partially-ordered-space) vector space (rather than merely a preordered vector space) while others only require that it be a preordered vector space.\nWe will henceforth assume that every Riesz space and every vector lattice is an [ordered vector space](https://bliptext.com/articles/ordered-vector-space) but that a preordered vector lattice is not necessarily partially ordered.\nIf E is an ordered vector space over \\R whose positive cone C (the elements \\,\\geq 0) is generating (that is, such that E = C - C), and if for every x, y \\in C either or exists, then E is a vector lattice.\n\n## Intervals\nAn **order interval** in a partially ordered vector space is a [convex set](https://bliptext.com/articles/convex-set) of the form\nIn an ordered real vector space, every interval of the form [-x, x] is [balanced](https://bliptext.com/articles/balanced-set).\nFrom axioms 1 and 2 above it follows that and implies\nA subset is said to be **order bounded** if it is contained in some order interval.\nAn **[order unit](https://bliptext.com/articles/order-unit)** of a preordered vector space is any element x such that the set [-x, x] is [absorbing](https://bliptext.com/articles/absorbing-set).\nThe set of all [linear functionals](https://bliptext.com/articles/linear-functional) on a preordered vector space V that map every order interval into a bounded set is called the **[order bound dual](https://bliptext.com/articles/order-bound-dual)** of V and denoted by V^b.\nIf a space is ordered then its order bound dual is a vector subspace of its [algebraic dual](https://bliptext.com/articles/algebraic-dual).\nA subset A of a vector lattice E is called ****[order complete](https://bliptext.com/articles/order-complete)**** if for every non-empty subset such that B is order bounded in A, both \\sup B and \\inf B exist and are elements of A. We say that a vector lattice E is ****[order complete](https://bliptext.com/articles/order-complete)**** if E is an ****[order complete](https://bliptext.com/articles/order-complete)**** subset of E.\n\n# Classification\nFinite-dimensional Riesz spaces are entirely classified by the [Archimedean property](https://bliptext.com/articles/archimedean-property):\nThe same result does not hold in infinite dimensions.\nFor an example due to [Kaplansky](https://bliptext.com/articles/irving-kaplansky), consider the vector space V of functions on that are continuous except at finitely many points, where they have a [pole](https://bliptext.com/articles/zeros-and-poles) of second order.\nThis space is lattice-ordered by the usual pointwise comparison, but cannot be written as\nκ\nfor any [cardinal](https://bliptext.com/articles/cardinal-number)\nκ\n. On the other hand, [epi-mono factorization](https://bliptext.com/articles/regular-category) in the category of\n-vector spaces also applies to Riesz spaces: every lattice-ordered vector space [injects](https://bliptext.com/articles/injective-function) into a quotient of\nκ\nby a [solid](https://bliptext.com/articles/solid-set) subspace.\n\n# Basic properties\nEvery Riesz space is a [partially ordered vector space](https://bliptext.com/articles/ordered-vector-space), but not every [partially ordered vector space](https://bliptext.com/articles/ordered-vector-space) is a Riesz space.\nNote that for any subset A of X, whenever either the supremum or infimum exists (in which case they both exist).\nIf x \\geq 0 and y \\geq 0 then\nFor all in a Riesz space X,\n\n## Absolute value\nFor every element x in a Riesz space X, the **absolute value** of x, denoted by |x|, is defined to be where this satisfies and |x| \\geq 0.\nFor any x, y \\in X and any real number r, we have and\n\n## Disjointness\nTwo elements in a vector lattice X are said to be **lattice **disjoint**** or **disjoint** if in which case we write x \\perp y. Two elements are **disjoint** if and only if If are **disjoint** then and where for any element z, and\nWe 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 A \\perp B. If A is the singleton set \\{ a \\} then we will write a \\perp B in place of For any set A, we define the **disjoint** complement to be the set\nDisjoint complements are always [bands](https://bliptext.com/articles/band-order-theory), but the converse is not true in general.\nIf A is a subset of X such that 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 \\{x\\}.\n\n## Representation as a disjoint sum of positive elements\nFor any x \\in X, let and where note that both of these elements are \\geq 0 and with\nThen x^+ and x^{-} are disjoint, and is the unique representation of x as the difference of disjoint elements that are \\geq 0.\nFor all x, y \\in X, and\nIf y \\geq 0 and x \\leq y then x^+ \\leq y. Moreover, x \\leq y if and only if and\nEvery Riesz space is a [distributive lattice](https://bliptext.com/articles/distributive-lattice); that is, it has the following equivalent properties: for all\nEvery Riesz space has the [Riesz decomposition property](https://bliptext.com/articles/approximately-finite-dimensional-c-algebra).\n\n# Order convergence\nThere 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.\nA sequence in a Riesz space E is said to **converge [monotone](https://bliptext.com/articles/monotone-sequence)ly** if it is a [monotone](https://bliptext.com/articles/monotone-sequence) decreasing (resp. increasing) sequence and its [infimum](https://bliptext.com/articles/infimum) (supremum) x exists in E and denoted (resp. ).\nA sequence in a Riesz space E is said to **converge in order** to x if there exists a monotone converging sequence in E such that\nIf u is a positive element of a Riesz space E then a sequence in E is said to **converge u-uniformly** to x if for any r > 0 there exists an N such that for all n > N.\n\n# Subspaces\nThe extra structure provided by these spaces provide for distinct kinds of Riesz subspaces.\nThe collection of each kind structure in a Riesz space (for example, the collection of all ideals) forms a [distributive lattice](https://bliptext.com/articles/distributive-lattice).\n\n## Sublattices\nIf X is a vector lattice then a **vector sublattice** is a vector subspace F of X such that for all x, y \\in F, belongs to F (where this supremum is taken in X).\nIt can happen that a subspace F of X is a vector lattice under its canonical order but is a vector sublattice of X.\n\n## Ideals\nA vector subspace I of a Riesz space E is called an **ideal** if it is **[solid](https://bliptext.com/articles/solid-set)**, meaning if for f \\in I and g \\in E, implies that g \\in I. The intersection of an arbitrary collection of **ideal**s 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**.\n\n## Bands and σ-Ideals\nA **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. \\sigma-Ideals are defined similarly, with the words 'arbitrary subset' replaced with 'countable subset'.\nClearly every band is a \\sigma-ideal, but the converse is not true in general.\nThe intersection of an arbitrary family of bands is again a band.\nAs with ideals, for every non-empty subset A of E, there exists a smallest band containing that subset, called\nA band generated by a singleton is called a **principal band**.\n\n## Projection bands\nA band B in a Riesz space, is called a **projection band**, if meaning every element f \\in E can be written uniquely as a sum of two elements, f = u + v with u \\in B and\nThere then also exists a positive linear idempotent, or, such that P_B(f) = u.\nThe collection of all projection bands in a Riesz space forms a [Boolean algebra](https://bliptext.com/articles/boolean-algebra).\nSome spaces do not have non-trivial projection bands (for example, C([0, 1])), so this Boolean algebra may be trivial.\n\n## Completeness\nA vector lattice is **complete** if every subset has both a supremum and an infimum.\nA vector lattice is **[Dedekind complete](https://bliptext.com/articles/dedekind-complete)** if each set with an upper bound has a supremum and each set with a lower bound has an infimum.\nAn order complete, regularly ordered vector lattice whose canonical image in its [order bidual](https://bliptext.com/articles/order-bidual) is order complete is called **minimal** and is said to be of **minimal** type.\n\n# Subspaces, quotients, and products\n**Sublattices**\nIf M is a vector subspace of a preordered vector space X then the canonical ordering on M induced by X's positive cone C is the preorder induced by the pointed convex cone C \\cap M, where this cone is proper if C is proper (that is, if ).\nA **sublattice** of a vector lattice X is a vector subspace M of X such that for all x, y \\in M, belongs to X (importantly, note that this supremum is taken in X and not in M).\nIf with 0 < p < 1, then the 2-dimensional vector subspace M of X defined by all maps of the form (where a, b \\in \\R) is a vector lattice under the induced order but is a sublattice of X. This despite X being an [order complete](https://bliptext.com/articles/order-complete) [Archimedean ordered](https://bliptext.com/articles/archimedean-order) [topological vector lattice](https://bliptext.com/articles/topological-vector-lattice).\nFurthermore, there exist vector a vector sublattice N of this space X such that N \\cap C has empty interior in X but no positive linear functional on N can be extended to a positive linear functional on X.\n**Quotient lattices**\nLet M be a vector subspace of an ordered vector space X having positive cone C, let be the canonical projection, and let\nThen \\hat{C} is a cone in X / M that induces a canonical preordering on the [quotient space](https://bliptext.com/articles/quotient-space-linear-algebra) X / M. If \\hat{C} is a proper cone in X / M then \\hat{C} makes X / M into an ordered vector space.\nIf M is [C-saturated](https://bliptext.com/articles/cone-saturated) then \\hat{C} defines the canonical order of X / M. Note that X=\\R^2_{0} provides an example of an ordered vector space where \\pi(C) is not a proper cone.\nIf X is a vector lattice and N is a [solid](https://bliptext.com/articles/solid-set) vector subspace of X then \\hat{C} defines the canonical order of X / M under which L / M is a vector lattice and the canonical map is a vector lattice homomorphism.\nFurthermore, if X is [order complete](https://bliptext.com/articles/order-complete) and M is a band in X then X / M is isomorphic with M^{\\bot}.\nAlso, if M is solid then the [order topology](https://bliptext.com/articles/order-topology-functional-analysis) of X / M is the quotient of the [order topology](https://bliptext.com/articles/order-topology-functional-analysis) on X.\nIf X is a [topological vector lattice](https://bliptext.com/articles/topological-vector-lattice) and M is a closed [solid](https://bliptext.com/articles/solid-set) sublattice of X then X / L is also a [topological vector lattice](https://bliptext.com/articles/topological-vector-lattice).\n**Product**\nIf S is any set then the space X^S of all functions from S into X is canonically ordered by the proper cone\nSuppose that is a family of preordered vector spaces and that the positive cone of X_\\alpha is C_\\alpha.\nThen is a pointed convex cone in which determines a canonical ordering on ;\nC is a proper cone if all C_\\alpha are proper cones.\n**Algebraic direct sum**\nThe algebraic [direct sum](https://bliptext.com/articles/direct-sum) of is a vector subspace of that is given the canonical subspace ordering inherited from\nIf are ordered vector subspaces of an ordered vector space X then X is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of X onto (with the canonical product order) is an [order isomorphism](https://bliptext.com/articles/order-isomorphism).\n\n# Spaces of linear maps\nA cone C in a vector space X is said to be **generating** if C - C is equal to the whole vector space.\nIf X and W are two non-trivial ordered vector spaces with respective positive cones P and Q, then P is generating in X if and only if the set is a proper cone in which is the space of all linear maps from X into W. In this case the ordering defined by C is called the ****canonical ordering**** of More generally, if M is any vector subspace of such that C \\cap M is a proper cone, the ordering defined by C \\cap M is called the ****canonical ordering**** of M.\nA linear map u : X \\to Y between two preordered vector spaces X and Y with respective **positive** cones C and D is called **positive** if If X and Y are vector lattices with Y [order complete](https://bliptext.com/articles/order-complete) and if H is the set of all **positive** linear maps from X into Y then the subspace M := H - H of is an [order complete](https://bliptext.com/articles/order-complete) vector lattice under its canonical order;\nfurthermore, M contains exactly those linear maps that map order intervals of X into order intervals of Y.\n\n## Positive functionals and the order dual\nA linear function f on a preordered vector space is called **positive** if x \\geq 0 implies\nThe set of all positive linear forms on a vector space, denoted by C^*, is a cone equal to the [polar](https://bliptext.com/articles/polar-set) of - C. The **[order dual](https://bliptext.com/articles/order-dual-functional-analysis)** of an ordered vector space X is the set, denoted by X^+, defined by Although there do exist ordered vector spaces for which set equality does hold.\n\n# Vector lattice homomorphism\nSuppose that X and Y are preordered vector lattices with positive cones C and D and let u : X \\to Y be a map.\nThen u is a **preordered vector lattice homomorphism** if u is linear and if any one of the following equivalent conditions hold:\n

u preserves the lattice operations
for all x, y \\in X.
for all x, y \\in X.
for all x \\in X.
for all x \\in X.
u(C) = D and u^{-1}(0) is a [solid](https://bliptext.com/articles/solid-set) subset of X.
if x \\geq 0 then
u is order preserving.

\nA pre-ordered vector lattice homomorphism that is bijective is a **pre-ordered vector lattice isomorphism**.\nA pre-ordered **vector lattice homomorphism** between two Riesz spaces is called a **vector lattice homomorphism**;\nif it is also bijective, then it is called a **vector lattice isomorphism**.\nIf u is a non-zero linear functional on a vector lattice X with positive cone C then the following are equivalent:\n

is a surjective vector lattice homomorphism.
for all x \\in X.
u \\geq 0 and u^{-1}(0) is a [solid](https://bliptext.com/articles/solid-set) hyperplane in X.
u' generates an extreme ray of the cone X^* in X^*.\n

\nAn **extreme ray** of the cone C is a set where x \\in C, x is non-zero, and if y \\in C is such that x - y \\in C then y = s x for some s such that\nA vector lattice homomorphism from X into Y is a [topological homomorphism](https://bliptext.com/articles/topological-homomorphism) when X and Y are given their respective [order topologies](https://bliptext.com/articles/order-topology-functional-analysis).\n\n# Projection properties\nThere are numerous projection properties that Riesz spaces may have.\nA Riesz space is said to have the (principal) projection property if every (principal) band is a projection band.\nThe so-called **main inclusion theorem** relates the following additional properties to the (principal) projection property: A Riesz space is...\nThen these properties are related as follows.\nSDC implies DC; DC implies both Dedekind \\sigma-completeness and the projection property; Both Dedekind \\sigma-completeness and the projection property separately imply the principal projection property; and the principal projection property implies the [Archimedean property](https://bliptext.com/articles/archimedean-property).\nNone of the reverse implications hold, but Dedekind \\sigma-completeness and the projection property together imply DC.\n\n# Examples\n\n# Properties","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T08:49:19.472Z","updated_at":"2025-01-26T08:49:19.472Z","search_vector":"'riesz':1 'space':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **Riesz space**, **[lattice](https://bliptext.com/articles/lattice-order)-ordered vector space** or **vector [lattice](https://bliptext.com/articles/lattice-order)** is a [partially ordered vector space](https://bliptext.com/articles/ordered-vector-space) where the [order structure](https://bliptext.com/articles/order-structure) is a [lattice](https://bliptext.com/articles/lattice-order).\nRiesz spaces are named after [Frigyes Riesz](https://bliptext.com/articles/frigyes-riesz) who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires.\nRiesz spaces have wide-ranging applications.\nThey are important in [measure theory](https://bliptext.com/articles/measure-theory), in that important results are special cases of results for Riesz spaces.\nFor example, the [Radon–Nikodym theorem](https://bliptext.com/articles/radon-nikodym-theorem) follows as a special case of the [Freudenthal spectral theorem](https://bliptext.com/articles/freudenthal-spectral-theorem).\nRiesz spaces have also seen application in [mathematical economics](https://bliptext.com/articles/mathematical-economics) through the work of Greek-American economist and mathematician [Charalambos D. Aliprantis](https://bliptext.com/articles/charalambos-d-aliprantis).\n\n# Definition\n\n## Preliminaries\nIf X is an [ordered vector space](https://bliptext.com/articles/ordered-vector-space) (which by definition is a vector space over the [reals](https://bliptext.com/articles/real-number)) and if S is a subset of X then an element b \\in X is an **upper bound** (resp. **lower bound**) of S if s \\leq b (resp. s \\geq b) for all s \\in S. An element a in X is the least **upper bound** or **[supremum](https://bliptext.com/articles/supremum)** (resp. greater **lower bound** or **[infimum](https://bliptext.com/articles/infimum)**) of S if it is an **upper bound** (resp. a **lower bound**) of S and if for any **upper bound** (resp. any **lower bound**) b of S, a \\leq b (resp. a \\geq b).\n\n## Definitions\n\n## Preordered vector lattice\nA **preordered vector lattice** is a pre[ordered vector space](https://bliptext.com/articles/ordered-vector-space) E in which every pair of elements has a [supremum](https://bliptext.com/articles/supremum).\nMore explicitly, a **[preorder](https://bliptext.com/articles/preorder)ed vector lattice** is vector space endowed with a [preorder](https://bliptext.com/articles/preorder), \\,\\leq,\\, such that for any :\nThe preorder, together with items 1 and 2, which make it \"compatible with the vector space structure\", make E a preordered vector space.\nItem 3 says that the preorder is a [join semilattice](https://bliptext.com/articles/join-semilattice).\nBecause the preorder is compatible with the vector space structure, one can show that any pair also have an [infimum](https://bliptext.com/articles/infimum), making E also a [meet semilattice](https://bliptext.com/articles/meet-semilattice), hence a lattice.\nA preordered vector space E is a preordered vector lattice if and only if it satisfies any of the following equivalent properties:\n \n For any x, y \\in E, their [supremum](https://bliptext.com/articles/supremum) exists in E. \n For any x, y \\in E, their [infimum](https://bliptext.com/articles/infimum) exists in E. \n For any x, y \\in E, their infimum and their supremum exist in E. \n For any x \\in E, exists in E. \n \n\n## Riesz space and vector lattices\nA **Riesz space** or a **vector lattice** is a preordered **vector lattice** whose preorder is a [partial order](https://bliptext.com/articles/partial-order).\nEquivalently, it is an [ordered vector space](https://bliptext.com/articles/ordered-vector-space)\nfor which the ordering is a [lattice](https://bliptext.com/articles/lattice-order).\nNote that many authors required that a vector lattice be a [partially ordered](https://bliptext.com/articles/partially-ordered-space) vector space (rather than merely a preordered vector space) while others only require that it be a preordered vector space.\nWe will henceforth assume that every Riesz space and every vector lattice is an [ordered vector space](https://bliptext.com/articles/ordered-vector-space) but that a preordered vector lattice is not necessarily partially ordered.\nIf E is an ordered vector space over \\R whose positive cone C (the elements \\,\\geq 0) is generating (that is, such that E = C - C), and if for every x, y \\in C either or exists, then E is a vector lattice.\n\n## Intervals\nAn **order interval** in a partially ordered vector space is a [convex set](https://bliptext.com/articles/convex-set) of the form\nIn an ordered real vector space, every interval of the form [-x, x] is [balanced](https://bliptext.com/articles/balanced-set).\nFrom axioms 1 and 2 above it follows that and implies\nA subset is said to be **order bounded** if it is contained in some order interval.\nAn **[order unit](https://bliptext.com/articles/order-unit)** of a preordered vector space is any element x such that the set [-x, x] is [absorbing](https://bliptext.com/articles/absorbing-set).\nThe set of all [linear functionals](https://bliptext.com/articles/linear-functional) on a preordered vector space V that map every order interval into a bounded set is called the **[order bound dual](https://bliptext.com/articles/order-bound-dual)** of V and denoted by V^b.\nIf a space is ordered then its order bound dual is a vector subspace of its [algebraic dual](https://bliptext.com/articles/algebraic-dual).\nA subset A of a vector lattice E is called ****[order complete](https://bliptext.com/articles/order-complete)**** if for every non-empty subset such that B is order bounded in A, both \\sup B and \\inf B exist and are elements of A. We say that a vector lattice E is ****[order complete](https://bliptext.com/articles/order-complete)**** if E is an ****[order complete](https://bliptext.com/articles/order-complete)**** subset of E.\n\n# Classification\nFinite-dimensional Riesz spaces are entirely classified by the [Archimedean property](https://bliptext.com/articles/archimedean-property):\nThe same result does not hold in infinite dimensions.\nFor an example due to [Kaplansky](https://bliptext.com/articles/irving-kaplansky), consider the vector space V of functions on that are continuous except at finitely many points, where they have a [pole](https://bliptext.com/articles/zeros-and-poles) of second order.\nThis space is lattice-ordered by the usual pointwise comparison, but cannot be written as\nκ\nfor any [cardinal](https://bliptext.com/articles/cardinal-number)\nκ\n. On the other hand, [epi-mono factorization](https://bliptext.com/articles/regular-category) in the category of\n-vector spaces also applies to Riesz spaces: every lattice-ordered vector space [injects](https://bliptext.com/articles/injective-function) into a quotient of\nκ\nby a [solid](https://bliptext.com/articles/solid-set) subspace.\n\n# Basic properties\nEvery Riesz space is a [partially ordered vector space](https://bliptext.com/articles/ordered-vector-space), but not every [partially ordered vector space](https://bliptext.com/articles/ordered-vector-space) is a Riesz space.\nNote that for any subset A of X, whenever either the supremum or infimum exists (in which case they both exist).\nIf x \\geq 0 and y \\geq 0 then\nFor all in a Riesz space X,\n\n## Absolute value\nFor every element x in a Riesz space X, the **absolute value** of x, denoted by |x|, is defined to be where this satisfies and |x| \\geq 0.\nFor any x, y \\in X and any real number r, we have and\n\n## Disjointness\nTwo elements in a vector lattice X are said to be **lattice **disjoint**** or **disjoint** if in which case we write x \\perp y. Two elements are **disjoint** if and only if If are **disjoint** then and where for any element z, and\nWe 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 A \\perp B. If A is the singleton set \\{ a \\} then we will write a \\perp B in place of For any set A, we define the **disjoint** complement to be the set\nDisjoint complements are always [bands](https://bliptext.com/articles/band-order-theory), but the converse is not true in general.\nIf A is a subset of X such that 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 \\{x\\}.\n\n## Representation as a disjoint sum of positive elements\nFor any x \\in X, let and where note that both of these elements are \\geq 0 and with\nThen x^+ and x^{-} are disjoint, and is the unique representation of x as the difference of disjoint elements that are \\geq 0.\nFor all x, y \\in X, and\nIf y \\geq 0 and x \\leq y then x^+ \\leq y. Moreover, x \\leq y if and only if and\nEvery Riesz space is a [distributive lattice](https://bliptext.com/articles/distributive-lattice); that is, it has the following equivalent properties: for all\nEvery Riesz space has the [Riesz decomposition property](https://bliptext.com/articles/approximately-finite-dimensional-c-algebra).\n\n# Order convergence\nThere 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.\nA sequence in a Riesz space E is said to **converge [monotone](https://bliptext.com/articles/monotone-sequence)ly** if it is a [monotone](https://bliptext.com/articles/monotone-sequence) decreasing (resp. increasing) sequence and its [infimum](https://bliptext.com/articles/infimum) (supremum) x exists in E and denoted (resp. ).\nA sequence in a Riesz space E is said to **converge in order** to x if there exists a monotone converging sequence in E such that\nIf u is a positive element of a Riesz space E then a sequence in E is said to **converge u-uniformly** to x if for any r > 0 there exists an N such that for all n > N.\n\n# Subspaces\nThe extra structure provided by these spaces provide for distinct kinds of Riesz subspaces.\nThe collection of each kind structure in a Riesz space (for example, the collection of all ideals) forms a [distributive lattice](https://bliptext.com/articles/distributive-lattice).\n\n## Sublattices\nIf X is a vector lattice then a **vector sublattice** is a vector subspace F of X such that for all x, y \\in F, belongs to F (where this supremum is taken in X).\nIt can happen that a subspace F of X is a vector lattice under its canonical order but is a vector sublattice of X.\n\n## Ideals\nA vector subspace I of a Riesz space E is called an **ideal** if it is **[solid](https://bliptext.com/articles/solid-set)**, meaning if for f \\in I and g \\in E, implies that g \\in I. The intersection of an arbitrary collection of **ideal**s 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**.\n\n## Bands and σ-Ideals\nA **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. \\sigma-Ideals are defined similarly, with the words 'arbitrary subset' replaced with 'countable subset'.\nClearly every band is a \\sigma-ideal, but the converse is not true in general.\nThe intersection of an arbitrary family of bands is again a band.\nAs with ideals, for every non-empty subset A of E, there exists a smallest band containing that subset, called\nA band generated by a singleton is called a **principal band**.\n\n## Projection bands\nA band B in a Riesz space, is called a **projection band**, if meaning every element f \\in E can be written uniquely as a sum of two elements, f = u + v with u \\in B and\nThere then also exists a positive linear idempotent, or, such that P_B(f) = u.\nThe collection of all projection bands in a Riesz space forms a [Boolean algebra](https://bliptext.com/articles/boolean-algebra).\nSome spaces do not have non-trivial projection bands (for example, C([0, 1])), so this Boolean algebra may be trivial.\n\n## Completeness\nA vector lattice is **complete** if every subset has both a supremum and an infimum.\nA vector lattice is **[Dedekind complete](https://bliptext.com/articles/dedekind-complete)** if each set with an upper bound has a supremum and each set with a lower bound has an infimum.\nAn order complete, regularly ordered vector lattice whose canonical image in its [order bidual](https://bliptext.com/articles/order-bidual) is order complete is called **minimal** and is said to be of **minimal** type.\n\n# Subspaces, quotients, and products\n**Sublattices**\nIf M is a vector subspace of a preordered vector space X then the canonical ordering on M induced by X's positive cone C is the preorder induced by the pointed convex cone C \\cap M, where this cone is proper if C is proper (that is, if ).\nA **sublattice** of a vector lattice X is a vector subspace M of X such that for all x, y \\in M, belongs to X (importantly, note that this supremum is taken in X and not in M).\nIf with 0 \n u preserves the lattice operations \n for all x, y \\in X. \n for all x, y \\in X. \n for all x \\in X. \n for all x \\in X. \n u(C) = D and u^{-1}(0) is a [solid](https://bliptext.com/articles/solid-set) subset of X. \n if x \\geq 0 then \n u is order preserving. \n \nA pre-ordered vector lattice homomorphism that is bijective is a **pre-ordered vector lattice isomorphism**.\nA pre-ordered **vector lattice homomorphism** between two Riesz spaces is called a **vector lattice homomorphism**;\nif it is also bijective, then it is called a **vector lattice isomorphism**.\nIf u is a non-zero linear functional on a vector lattice X with positive cone C then the following are equivalent:\n \n is a surjective vector lattice homomorphism. \n for all x \\in X. \n u \\geq 0 and u^{-1}(0) is a [solid](https://bliptext.com/articles/solid-set) hyperplane in X. \n u' generates an extreme ray of the cone X^* in X^*.\n \nAn **extreme ray** of the cone C is a set where x \\in C, x is non-zero, and if y \\in C is such that x - y \\in C then y = s x for some s such that\nA vector lattice homomorphism from X into Y is a [topological homomorphism](https://bliptext.com/articles/topological-homomorphism) when X and Y are given their respective [order topologies](https://bliptext.com/articles/order-topology-functional-analysis).\n\n# Projection properties\nThere are numerous projection properties that Riesz spaces may have.\nA Riesz space is said to have the (principal) projection property if every (principal) band is a projection band.\nThe so-called **main inclusion theorem** relates the following additional properties to the (principal) projection property: A Riesz space is...\nThen these properties are related as follows.\nSDC implies DC; DC implies both Dedekind \\sigma-completeness and the projection property; Both Dedekind \\sigma-completeness and the projection property separately imply the principal projection property; and the principal projection property implies the [Archimedean property](https://bliptext.com/articles/archimedean-property).\nNone of the reverse implications hold, but Dedekind \\sigma-completeness and the projection property together imply DC.\n\n# Examples\n\n# Properties","revisionCount":0},"hasMore":false,"nextCursor":null}

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