Partially ordered vector space, ordered as a lattice
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
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 ![{\displaystyle x+z\leq y+z.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- Positive Homogeneity: For any scalar
implies ![{\displaystyle ax\leq ay.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- For any pair of vectors
there exists a supremum (denoted
) in
with respect to the order ![{\displaystyle \,(\leq ).\,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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:
- For any
their supremum exists in ![{\displaystyle E.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- For any
their infimum exists in ![{\displaystyle E.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- For any
their infimum and their supremum exist in ![{\displaystyle E.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- For any
exists in
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 spacefor 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.
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.
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.
An order unit of a preordered vector space is any element
such that the set
is absorbing.
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
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 ![{\displaystyle E.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Classification
Finite-dimensional Riesz spaces are entirely classified by the Archimedean property:
- Theorem: 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. 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]
Basic properties
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
of
whenever either the supremum or infimum exists (in which case they both exist).If
and
then
For all
in a Riesz space
![{\displaystyle a-\inf(x,y)+b=\sup(a-x+b,a-y+b).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Absolute value
For every element
in a Riesz space
the absolute value of
denoted by
is defined to be
where this satisfies
and
For any
and any real number
we have
and ![{\displaystyle |x+y|\leq |x|+|y|.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Disjointness
Two elements
in a vector lattice
are said to be lattice disjoint or disjoint if
in which case we write
Two elements
are disjoint if and only if
If
are disjoint then
and
where for any element
and
We say that two sets
and
are disjoint if
and
are disjoint for all
and all
in which case we write
If
is the singleton set
then we will write
in place of
For any set
we define the disjoint complement to be the set
Disjoint complements are always bands, but the converse is not true in general.
If
is a subset of
such that
exists, and if
is a subset lattice in
that is disjoint from
then
is a lattice disjoint from ![{\displaystyle \{x\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Representation as a disjoint sum of positive elements
For any
let
and
where note that both of these elements are
and
with
Then
and
are disjoint, and
is the unique representation of
as the difference of disjoint elements that are
For all
and
If
and
then
Moreover,
if and only if
and ![{\displaystyle x^{-}\leq y^{-}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Every Riesz space is a distributive lattice; that is, it has the following equivalent[Note 1] properties:[8] for all ![{\displaystyle x,y,z\in X}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle (x\wedge y)\vee (y\wedge z)\vee (z\wedge x)=(x\vee y)\wedge (y\vee z)\wedge (z\vee x).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
always imply ![{\displaystyle x=y.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Every Riesz space has the Riesz decomposition property.
Order convergence
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
in a Riesz space
is said to converge monotonely if it is a monotone decreasing (resp. increasing) sequence and its infimum (supremum)
exists in
and denoted
(resp.
).
A sequence
in a Riesz space
is said to converge in order to
if there exists a monotone converging sequence
in
such that ![{\displaystyle \left|x_{n}-x\right|<p_{n}\downarrow 0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
If
is a positive element of a Riesz space
then a sequence
in
is said to converge u-uniformly to
if for any
there exists an
such that
for all ![{\displaystyle n>N.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Subspaces
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
is a vector lattice then a vector sublattice is a vector subspace
of
such that for all
belongs to
(where this supremum is taken in
).
It can happen that a subspace
of
is a vector lattice under its canonical order but is not a vector sublattice of ![{\displaystyle X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Ideals
A vector subspace
of a Riesz space
is called an ideal if it is solid, meaning if for
and
implies that
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
of
and is called the ideal generated by
An Ideal generated by a singleton is called a principal ideal.
Bands and σ-Ideals
A band
in a Riesz space
is defined to be an ideal with the extra property, that for any element
for which its absolute value
is the supremum of an arbitrary subset of positive elements in
that
is actually in
-Ideals are defined similarly, with the words 'arbitrary subset' replaced with 'countable subset'. Clearly every band is a
-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
of
there exists a smallest band containing that subset, called the band generated by
A band generated by a singleton is called a principal band.
Projection bands
A band
in a Riesz space, is called a projection band, if
meaning every element
can be written uniquely as a sum of two elements,
with
and
There then also exists a positive linear idempotent, or projection,
such that ![{\displaystyle P_{B}(f)=u.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The collection of all projection bands in a Riesz space forms a Boolean algebra. Some spaces do not have non-trivial projection bands (for example,
), so this Boolean algebra may be trivial.
Completeness
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.
Subspaces, quotients, and products
Sublattices
If
is a vector subspace of a preordered vector space
then the canonical ordering on
induced by
's positive cone
is the preorder induced by the pointed convex cone
where this cone is proper if
is proper (that is, if
).
A sublattice of a vector lattice
is a vector subspace
of
such that for all
belongs to
(importantly, note that this supremum is taken in
and not in
).
If
with
then the 2-dimensional vector subspace
of
defined by all maps of the form
(where
) is a vector lattice under the induced order but is not a sublattice of
This despite
being an order complete Archimedean ordered topological vector lattice.
Furthermore, there exist vector a vector sublattice
of this space
such that
has empty interior in
but no positive linear functional on
can be extended to a positive linear functional on ![{\displaystyle X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Quotient lattices
Let
be a vector subspace of an ordered vector space
having positive cone
let
be the canonical projection, and let
Then
is a cone in
that induces a canonical preordering on the quotient space
If
is a proper cone in
then
makes
into an ordered vector space.
If
is
-saturated then
defines the canonical order of
Note that
provides an example of an ordered vector space where
is not a proper cone.
If
is a vector lattice and
is a solid vector subspace of
then
defines the canonical order of
under which
is a vector lattice and the canonical map
is a vector lattice homomorphism.
Furthermore, if
is order complete and
is a band in
then
is isomorphic with
Also, if
is solid then the order topology of
is the quotient of the order topology on ![{\displaystyle X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
If
is a topological vector lattice and
is a closed solid sublattice of
then
is also a topological vector lattice.
Product
If
is any set then the space
of all functions from
into
is canonically ordered by the proper cone ![{\displaystyle \left\{f\in X^{S}:f(s)\in C{\text{ for all }}s\in S\right\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Suppose that
is a family of preordered vector spaces and that the positive cone of
is
Then
is a pointed convex cone in
which determines a canonical ordering on
;
is a proper cone if all
are proper cones.
Algebraic direct sum
The algebraic direct sum
of
is a vector subspace of
that is given the canonical subspace ordering inherited from
If
are ordered vector subspaces of an ordered vector space
then
is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of
onto
(with the canonical product order) is an order isomorphism.
Spaces of linear maps
A cone
in a vector space
is said to be generating if
is equal to the whole vector space.
If
and
are two non-trivial ordered vector spaces with respective positive cones
and
then
is generating in
if and only if the set
is a proper cone in
which is the space of all linear maps from
into
In this case the ordering defined by
is called the canonical ordering of
More generally, if
is any vector subspace of
such that
is a proper cone, the ordering defined by
is called the canonical ordering of ![{\displaystyle M.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A linear map
between two preordered vector spaces
and
with respective positive cones
and
is called positive if
If
and
are vector lattices with
order complete and if
is the set of all positive linear maps from
into
then the subspace
of
is an order complete vector lattice under its canonical order;
furthermore,
contains exactly those linear maps that map order intervals of
into order intervals of ![{\displaystyle Y.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Positive functionals and the order dual
A linear function
on a preordered vector space is called positive if
implies
The set of all positive linear forms on a vector space, denoted by
is a cone equal to the polar of
The order dual of an ordered vector space
is the set, denoted by
defined by
Although
there do exist ordered vector spaces for which set equality does not hold.
Vector lattice homomorphism
Suppose that
and
are preordered vector lattices with positive cones
and
and let
be a map.
Then
is a preordered vector lattice homomorphism if
is linear and if any one of the following equivalent conditions hold:
preserves the lattice operations
for all ![{\displaystyle x,y\in X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for all ![{\displaystyle x,y\in X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for all ![{\displaystyle x\in X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for all ![{\displaystyle x\in X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
is a solid subset of ![{\displaystyle X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- if
then ![{\displaystyle u(x)\geq 0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is order preserving.
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
is a non-zero linear functional on a vector lattice
with positive cone
then the following are equivalent:
is a surjective vector lattice homomorphism.
for all ![{\displaystyle x\in X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
is a solid hyperplane in ![{\displaystyle X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
generates an extreme ray of the cone
in ![{\displaystyle X^{*}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
An extreme ray of the cone
is a set
where
is non-zero, and if
is such that
then
for some
such that ![{\displaystyle 0\leq s\leq 1.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A vector lattice homomorphism from
into
is a topological homomorphism when
and
are given their respective order topologies.
Projection properties
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
-complete if every countable nonempty set, bounded above, has a supremum; and - Archimedean property if, for every pair of positive elements
and
, whenever the inequality
holds for all integers
,
.
Then these properties are related as follows. SDC implies DC; DC implies both Dedekind
-completeness and the projection property; Both Dedekind
-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
-completeness and the projection property together imply DC.
Examples
- The space of continuous real valued functions with compact support on a topological space
with the pointwise partial order defined by
when
for all
is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless
satisfies further conditions (for example, being extremally disconnected). - Any
space with the (almost everywhere) pointwise partial order is a Dedekind complete Riesz space. - The space
with the lexicographical order is a non-Archimedean Riesz space.
Properties
See also
Notes
- ^ The conditions are equivalent only when they apply to all triples in a lattice. There are elements in (for example) N5 that satisfy the first equation but not the second.
References
- ^ Fremlin, Measure Theory, claim 352L.
- ^ Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications (3rd ed.). American Mathematical Society. p. 11. ISBN 0-8218-1025-1. §6, Theorem 9
- ^ Luxemburg, W.A.J.; Zaanen, A.C. (1971). Riesz Spaces : Vol. 1. London: North Holland. pp. 122–138. ISBN 0720424518. Retrieved 8 January 2018.
Bibliography
- 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
External links