Riesz space

vector lattice

A real partially ordered vector space $ X $( cf. Partially ordered set; Vector space) in which

1) the vector space structure and the partial order are compatible. i.e. from $ x, y, z \in X $ and $ x < y $ follows that $ x+ z < y+ z $ and from $ x \in X $, $ x > 0 $, $ \lambda \in \mathbf R $, $ \lambda > 0 $ follows $ \lambda x > 0 $;

2) for any two elements $ x, y $ there exists $ \sup ( x, y) \in X $. In particular, the supremum and infimum of any finite set exist.

In Soviet scientific literature Riesz spaces are usually called $ K $- lineals. Such spaces were first introduced by F. Riesz in 1928.

The space $ C[ a, b] $ of real continuous functions with the pointwise order is an example of a Riesz space. For any element $ x $ of a Riesz space one can define $ x _ {+} = \sup ( x, 0) $, $ x _ {-} = \sup (- x, 0) $ and $ | x | = x _ {+} + x _ {-} $. It turns out that $ x = x _ {+} - x _ {-} $. In Riesz spaces one can introduce two types of convergence of a sequence $ \{ x _ {n} \} $. Order convergence, $ o $- convergence: $ x _ {n} \rightarrow ^ {o} x _ {0} $ if there exist a monotone increasing sequence $ \{ y _ {n} \} $ and a monotone decreasing sequence $ \{ z _ {n} \} $ such that $ y _ {n} \leq x _ {n} \leq z _ {n} $ and $ \sup y _ {n} = \inf z _ {n} = x _ {0} $. Relative uniform convergence, $ r $- convergence: $ x _ {n} \rightarrow ^ {r} x _ {0} $ if there exists an element $ u > 0 $ such that for any $ \epsilon > 0 $ there exists an $ n _ {0} $ such that $ | x _ {n} - x _ {0} | < \epsilon u $ for $ n \geq n _ {0} $( $ r $- convergence is also called convergence with a regulator). The concepts of $ o $- and $ r $- convergence have many of the usual properties of convergence of numerical sequences and can be naturally generalized to nets $ \{ x _ \alpha \} _ {\alpha \in \mathfrak A } \subset X $.

A Riesz space is called Archimedean if $ x, y \in X $ and $ nx \leq y $ for $ n = 1, 2 , . . . $ imply $ x \leq 0 $. In Archimedean Riesz spaces, $ \lambda _ {n} \rightarrow \lambda _ {0} $ and $ x _ {n} \rightarrow ^ {o} x _ {0} $ imply $ \lambda _ {n} x _ {n} \rightarrow ^ {o} \lambda _ {0} x _ {0} $( $ \lambda _ {n} , \lambda _ {0} \in \mathbf R $, $ x _ {n} , x _ {0} \in X $), and $ r $- convergence implies $ o $- convergence.

References

[1]	F. Riesz, "Sur la décomposition des opérations fonctionelles linéaires" , Atti congress. internaz. mathematici (Bologna, 1928) , 3 , Zanichelli (1930) pp. 143–148
[2]	W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)
[3]	B.Z. Vulikh, "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff (1967) (Translated from Russian)

Comments

A Riesz subspace of a Riesz space $ L $ is a linear subspace $ K $ of $ L $ such that $ \sup ( f, g) = f \lor g $ and $ \inf ( f, g) = f \wedge g $ are in $ K $ whenever $ f, g \in K $( where the sup and inf are those of $ L $). A subspace $ A $ of $ L $ that is an order ideal, i.e. $ f \in A $, $ g \in L $, $ | g | \leq | f | $ imply that $ g \in A $, is called a Riesz ideal. Such subspaces are called sublineals and normal sublineals in the Soviet literature. A band is a Riesz ideal $ A $ such that $ \sup D $ in $ A $ for $ D \subset A $ if $ \sup D $ exists in $ L $. A band is often called a component in the Soviet literature.

A linear operator $ T $ from a Riesz space $ L $ to a Riesz space $ M $ is called positive if $ Tf \geq 0 $ for all $ f \geq 0 $, $ f \in L $. A set $ D $ in $ L $ is called order bounded if there exist $ f, g \in L $ such that $ f \leq d \leq g $ for all $ d \in D $. The linear operator $ T $ is called order bounded if it takes order-bounded sets to order-bounded sets. Taking the positive operators as the positive cone defines an order structure on the space of order-bounded operators, turning it into a Dedekind-complete Riesz space (the Freudenthal–Kantorovich theorem). Recall that a lattice is Dedekind complete if every subset bounded from below (respectively above) has an inf (respectively sup). A positive operator is order bounded, and so are differences $ T _ {1} - T _ {2} $ of positive operators, which are called regular operators. If $ M $ is Dedekind complete, the converse holds: Every order-bounded operator $ T $ admits a Jordan decomposition $ T = T _ {1} - T _ {2} $ as a difference of two positive operators.

A norm $ \| \cdot \| $ on a Riesz space $ L $ is a Riesz norm if $ | f | \leq | g | $ implies $ \| f \| \leq \| g \| $. A Riesz semi-norm is a semi-norm with the same compatibility conditions. A Riesz space with a Riesz norm is a normed Riesz space. A norm-complete normed Riesz space is a Banach lattice. An order-bounded operator $ T $ from a Banach lattice $ L $ to a Dedekind-complete normed Riesz space is norm bounded.

Let $ T _ {b} ( L, M) $ be the space of order-bounded operators from a Riesz space $ L $ to a Dedekind-complete Riesz space $ M $. $ T \in T _ {b} ( L, M) $ is called sequentially order continuous, or $ \tau $- order continuous, if for every sequence $ u _ {n} \downarrow 0 $( i.e. that is monotonically decreasing to $ 0 $) it follows that $ \inf | Tu _ {n} | = 0 $; it is called order continuous if $ \inf | Tu _ \tau | = 0 $ for every downwards directed system $ u _ \tau \rightarrow 0 $ in $ L $( cf. Directed set). The Soviet terminology for order-continuous and sequentially order-continuous linear operators is o-linear and (o)-linear. The order-continuous and $ \tau $- order continuous operators are bands in $ T _ {b} ( L, M) $. The order dual of a Riesz space $ L $ is the space of order-bounded operators of $ L $ into $ \mathbf R $. The result that this order dual is Dedekind complete goes back to F. Riesz.

There is a second important concept of duality in Riesz space theory, reminiscent of both linear duality and the algebraic geometric duality: "ideals zero sets" , that is basic to scheme theory. It is called Baker–Benyon duality (see the volume with supplementary articles).

In the theory of linear topological spaces (cf. Topological vector space) the following criterion for boundedness of a set is used: A set B is bounded (in this theory) if and only if for every sequence $ ( x _ {n} ) _ {n} $, $ x _ {n} \in B $, and sequence of real numbers $ ( \lambda _ {n} ) _ {n} $ converging to zero, one has that $ ( \lambda _ {n} x _ {n} ) _ {n} $ converges to zero as $ n \rightarrow \infty $. The question arises whether order-bounded sets in a Riesz space can be characterized in this way, using instead order convergence of the $ ( \lambda _ {n} x _ {n} ) _ {n} $ to zero. For arbitrary Dedekind-complete Riesz spaces this need not be true. The Dedekind-complete Riesz spaces for which this criterion holds are called $ K ^ {+} $- spaces.

Let now $ L $ be a normed space and $ M $ a Dedekind-complete Riesz space. A linear operator $ U : L \rightarrow M $ is called bo-linear if $ x _ {n} \rightarrow x $ in norm implies $ U x _ {n} \rightarrow U x $ in order convergence. If $ M $ is a $ K ^ {+} $- space, then $ U : L \rightarrow M $ is bo-linear if and only if the image $ U( S) $ of the unit sphere $ S $ in $ L $ is order bounded. The element of $ M $ defined by

$$ | U | = \sup _ {\| x \| \leq 1 } | U x | $$

is then called an abstract norm for the operator $ U $.

There is a variety of Riesz space analogues of Hahn–Banach type extension and existence theorems. A selection follows. Let $ L $ be a normed space, $ E $ a linear subset of $ L $ and $ U: E \rightarrow M $ a bo-linear operator into a Dedekind-complete Riesz space $ M $. Suppose that $ U $ possesses an abstract norm. Then the operator $ U $ admits a bo-linear extension to all of $ L $ with the same abstract norm. This is one of the Kantorovich extension theorems. Another extension theorem for Riesz spaces, also due to B.Z. Kantorovich, concerns the extension of positive operators: Let $ X $ be a Riesz space and $ E $ a linear subset that majorizes $ X $, i.e. for every $ x \in X $ there is an $ e \in E $ such that $ | x | \leq e $. Let $ U : E \rightarrow M $ be a positive additive operator from $ E $ into a Dedekind-complete Riesz space $ Y $. Then there exists an additive and positive extension of $ U $ to all of $ X $. Using these and/or related extension theorems one can show that a positive linear functional on a Riesz subspace of a Riesz space $ L $ that is majorized by a Riesz semi-norm can be extended to a positive functional on all of $ L $, a result which in turn serves to discuss when the order dual of $ L $ is at least non-zero.

Examples of Riesz spaces are provided by spaces of real-valued functions on a topological space (possibly, extended real functions), where the order is defined pointwise. As in the case of, e.g., Banach algebras (cf. Banach algebra), where the Gel'fand representation provides an answer, one asks whether an arbitrary Riesz space can be seen as a space of real-valued functions on a suitable space (of ideals). The answer for Riesz spaces is given by the Yosida representation theorem and its relatives.

In the integration theory (of real functions) a basic role is played by such operations as $ f = f ^ {+} - f ^ {-} $, $ | f | = f ^ {+} + f ^ {-} $, where $ f ^ {+} ( x) = \max ( f ( x) , 0) $, $ f ^ {-} = \max (- f( x), 0) $, which makes at least potentially credible that Riesz spaces might provide a suitable abstract setting for integration theory. This is indeed the case in the form of the Freudenthal spectral theorem, which will be discussed below.

Let $ X $ be a lattice with zero, $ 0 $. Let $ Y $ be a non-empty subset of $ X $; the set of $ x \in X $ that are disjoint from $ Y $, i.e. $ x \wedge y = 0 $ for all $ y \in Y $, is called the disjoint complement of $ Y $ in $ X $ and is denoted by $ Y ^ {d} $. In a Riesz space $ L $ two elements $ f , g $ are called disjoint if $ | f | \wedge | g | = 0 $. (This agrees with the previous definition if $ f $ and $ g $ are both positive.)

Given a band $ A $ in a Riesz space $ L $, the disjoint complement $ A ^ {d} $ is also a band. If $ L $ is Dedekind complete, $ L = A \oplus A ^ {d} $. In general, a band $ A $ such that $ L= A \oplus A ^ {d} $ is called a projection band. A Riesz space is said to have the (principal) projection property if every (principal) band is a projection band. Thus, a Dedekind-complete Riesz space has the projection property and, a fortiori, the principal projection property.

Let $ L $ be a Riesz space with the principal projection property, let $ e $ be a non-zero positive element of $ L $ and let $ f $ be an element in the band generated by $ e $. Let $ u _ \alpha = \sup ( \alpha e - f, 0 ) $ for $ - \infty < \alpha < \infty $, and let $ p _ \alpha $ be the component of $ e $ in the band $ B _ \alpha $ generated by $ u _ \alpha $ under the decomposition $ L = B _ \alpha \oplus B _ \alpha ^ {d} $. The set $ ( p _ \alpha ) _ \alpha $ is called the spectral system of $ f $ with respect to $ e $. Now suppose there is a finite interval such that $ ae \leq f \leq ( b - \epsilon ) e $ for some $ \epsilon > 0 $. Then $ p _ \alpha = 0 $ for $ \alpha \leq a $ and $ p _ \alpha = e $ for $ \alpha \geq b $. For every partition $ \pi $: $ a = \alpha _ {0} < \alpha _ {1} < \dots < \alpha _ {n} = b $ of $ [ a , b] $ one forms the lower and upper sums

$$ s ( \pi , f ) = \ \sum_{k=1}^ { n } \alpha _ {k-1} ( p _ {\alpha _ {k} } - p _ {\alpha _ {k-1} } ), $$

$$ u ( \pi , f ) = \sum_{k=1}^ { n } \alpha _ {k} ( p _ {\alpha _ {k} } - p _ {\alpha _ {k-1} } ) . $$

One then has the following result in abstract integration theory, known as the Freudenthal spectral theorem. Let $ L $, $ e $, $ f $, $ a $, $ b $, $ \epsilon $ be as above. Then

$$ \sup _ \pi s ( \pi , f ) = \ f = \inf _ \pi u( \pi , f ). $$

In the case that $ L $ is a Riesz space of real-valued functions on a space (especially a subset of $ \mathbf R $) and $ e( x) = 1 $, this spectral theorem expresses approximation properties of functions in $ L $ by "step functions" . The Radon–Nikodým theorem in measure theory and the Poisson formula for bounded harmonic functions on an open disc are special cases of the spectral theorem. The Freudenthal spectral theorem was one of the starting points of Riesz space theory.

References

[a1]	A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)
[a2]	H.H. Schaefer, "Banach lattices and positive operators" , Springer (1974)
[a3]	E. de Jonge, A.C.M. van Rooy, "Introduction to Riesz spaces" , Tracts , 8 , Math. Centre (1977)
[a4]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[a5]	B.Z. Kantorovich, B.Z. Vulikh, A.G. Pinsker, "Functional analysis in partially ordered spaces" , Moscow (1950) (In Russian)
[a6]	H. Freudenthal, "Teilweise geordneten Moduln" Proc. Royal Acad. Sci. Amsterdam , 39 (1936) pp. 641–651
[a7]	H. Nakano, "Modern spectral theory" , Maruzen (1950)