If for every pair f,g in L the least upper bound or supremum (denoted by f V g) and the greatest lower bound or infimum (denoted by f A g) with respect to the order exist in L, then L is said to be a Riesz -space or a vector grid. The Riesz space L is called cr-Dede type complete if every nonempty at most countable subset of L bounded from above has a supremum. The Riesz space L is called super Dedekind complete if every non-empty bounded subset D of L has a supremum which is also the supremum of an at most countable subset of D. The following theorem gives some information about the above concepts. i) L is Dedekind complete if and only if for every indexed system {fa}.
It is easily verified that Dd is a band of L for every nonempty subgroup D of L. Given a nonempty subgroup D of L there exists a smaller ideal containing D, that is, the intersection of all ideals containing D. The family of all the ideals they contain. The Riesz space L has the projection property (respectively the principal projection property) if every band (respectively every principal band) is a projection band. In particular, A is said to be a dense order in L, or simply a dense order, if [A}= L. An ideal of Lis is called an almost dense order in L if Add = L, and A is called a superorder of dense if for every u E L + there exists a sequence { un} ~ A such that.
Let L be a given Riesz space. i) If L is Archimedean, for an ideal A ~ L to be order-tight it is necessary and sufficient that Ad = [8}o. The Riesz space L satisfies s• = clS for all S ~ L if and only if L has the diagonal gap property.
CHAPTER 2
Suppose p is a mapping from the Riesz space L to a Dedekind complete Riesz space M such that. If the ordered vector space (a('b,;e>+) is a Dedekind complete Riesz space, then Mis a Dedekind is a Riesz space. Let L be a Riesz space with a nonzero positive linear functional and let M be a Dedekind complete Riesz space.
Let Land M be two Riesz spaces with M Dedekind complete, and let A be an ideal of L. The ordered bounded transformation S from L to M is called an extension of T if S(u) = T(u) for all u in A, i.e. Let Land M be two Riesz spaces with M Dedekind complete, and let A be an ideal of L. Then T has a minimal positive extension T on the whole of L, in the sense that for any positive.
Let Land M be two given Riesz spaces with M Dedekind complete, and let A be an ideal of L. Assume that A~ L is a Riesz subspace of L such that for every f in L, there exists u in A such that that If I ~ u.
CHAPTER 3
If Lis is a super Dedekind perfect Riesz space and Mis is a Dedekind perfect Riesz space, then every integral of /b(L, M) is a normal integral, i.e. (/: b)n = ( i'.'b)c. A Riesz space L is said to have order convergence to Lis stable if f (o) > 0 implies A. Nakano proved that a Riesz space L has Dedekind handing if and only if Lis is Archirnedean and any two Dedekind cornplections Riesz lares are isomorphic (see [19], Theorem 30.
I must be the Dede-type c ampleti e of the Archimedean Riesz space L, and let M be a Dede-type complete Riesz space. From this observation and the fact that Mis Dedekind supplements, it follows that S(u) = inf{T(g): g EL+; u ~ g} exists in M for every u in L r. 3 that T can be extended to a linear map T1 on L. Some sufficient conditions for a Riesz space to satisfy one of the completeness properties are given in the following theorem.
Let L and M be two Riesz spaces and let TT be a normal Riesz homomorphism of L ~ M, i.e. 11 is a Riesz homo-. morphism that is also a normal integral. i) If Lis a-Dedekind is complete, then Mis a-Dedekind is complete. ii) If Lis Dedekind is complete, then M Dedekind is complete. iii). If Lis super Dedekind is complete, then M super Dedekind is complete. i) The proof can be found in Theorem 65. Since Lis is an a-Dedekind full Riesz space, it is particularly Archimedean, so we say that N~d= {NT} (see Theorem 1.6).
The proof of necessity in the above theorem does not depend on the assumption that Lis a-Dedekind satisfies. Let M be two Riesz given spaces with M super Dedekind complete, and let T be a strictly positive linear transformation from L to M. It follows in particular that Lis super Dedekind is complete if it is a-Dedekind full. ii) If T is a strictly positive integral of £b(L, M), then T is a normal integral. N~ is a complete Riesz Dedekind superspace and if T is an integral, then T bounded in N~ is a normal integral.
The proof follows from the previous theorem by observing that T restricted to N; is strictly positive and that N; itself is a Dedekind complete Riesz space.•. THE COMPONENTS OF AN ORDER BOUNDED TRANSFORMATION Given two Riesz spaces Land M with M Dedekind complete we denote by (J:'b)n = Cf.b(L, M) )n' (~b)c = tf b(L, M) )c the bands of the normal integrals and integrals of ~b(L, M), respectively. For any sequence e ~Un t ffl we have ff. i) the constraints on A of different elements in (j' b) are different, and c.
CHAPTER 4
The order vector space (E, K) is called a Riesz space if the smallest upper bound of any two elements of E exists in E. If, in addition, there exists a basis for the 'r-neighborhoods of zero consisting of fixed sets , (a subset of V of a Riesz space L is called . a solid, if As I ~ lg I and g EV f EV imply), then the topological Riesz space (L, 7') is called a locally fixed Riesz space. The following examples show that many other implications do not hold in general, even in locally solid Riesz spaces. i) Let L be the Riesz space of all real valued functions defined on an uncountable set X and such that for every f E L there exists a real number f(oo) such that given any e > 0, we have.
Let r be the locally solid topology generated by the norm p(f) =sup{ jf(x) I: x E X}. ii) Let L be the Riesz space of all real continuous functions defined on [O, l], i.e. iii) Let 0 < p < 1 and let L be the real vector space of all real valued Lebsegue measurable functions defined on [O, l], such that. family of sets {WF ~} which is a filter basis for a neighborhood system 'n, u. of the origin for a uniquely determined linear topology. We recall that a Riesz space L has a countable order basis if there exists an at most countable subset of L such that the band generated by this subset is the entire space. So p defines a non-Hausdorff locally fixed topology ,- on L, i.e. L, 'T) is a locally fixed Riesz space that, as mentioned earlier, is non-Archimedean.
For locally convex Riesz spaces Hausdorff, which are also locally strong, there is another characterization of property (A, ii). L, .,-) be a locally convex Hausdorff space, locally rigid Riesz. L, .,.) be a locally satisfying Riesz space condition of Hausdorff (A, ii) and let M be a complete Riesz space of super Dede type. if and only if, NT~ -r-ideal closed. Recall that a Riesz space L has the Egoroff property if. implies the existence of such a sequence {v J.
Let L be a Riesz space and suppose that L . admits a metrizable solid local topology ,-, which also satisfies the condition {A,ii). Let L be a Riesz space and let (M, 'T) be a locally rigid metrizable condition of the Riesz space (A, ii), which is also super Dedekind complete. It is known that not every Riesz subspace of a Riesz space with the Egoroff property has the Egoroff property.
The following theorem gives another sufficient condition that the Riesz space L has the Egoroff property. If A is an ordered ideal of a Riesz space L with the Egoroff property and if L has the diagonal gap property, then L has the Egoroff property. If every order bounded increasing 1"'-Cauchy sequence has a metrizable locally full Riesz space (L, ir) r-~.
Now suppose that a metrizable locally solid Riesz space (L, r) has the stated property and that (A, iii) holds.
CHAPTER 5
THEFATOUPROPERTIES
We recall that a Riesz space is called a normed Riesz space if there exists a norm p such that p(u) = p( I u I) for all u EL, and if. Also, the property (A, i) implies the sequential Fatou property and the property (A, ii) implies the Fatou property. i) Let L be the Riesz space of all continuous functions on [O, l],. Then p is a Riesz norm for L without the sequential Fatou property, as we can see by using an argument similar to that of Example 4. ii) Let L be the Riesz space of all real series that are ultimately constant are, that is, f = ff (n)} is in L if there exists a constant c and a.
An easy verification shows that L p satisfies the Fatou property and therefore also satisfies the sequential Fatou property. iii) Let L be the Riesz space of all bounded real values, Lebesgue. Using this and the Lebesgue-dominated convergence theorem, we easily see that L satisfies the Fatou sequential property. page Next we will show that the norm complement of a Riesz normed space with the Fatou (sequential) property is also a Riesz normed space with the Fatou (sequential) property.
Given an Archimedean Riesz space L, there exists a Dedekind-complete and universally complete Riesz space K (unique to a Riesz isomorphism) such that (i) K Las contains a Riesz subspace. Let (L, 'T) be a Hausdorff locally solid Riesz space satisfying the Fatou property. i) L is strictly order dense in L. Given a Riesz space Land a Riesz semi-norm p on L we say that p is a Fatou semi-norm if 0 ~ ua.
The following theorem is based on Theorem 5. L, 'T) is a Hausdorff locally solid, locally convex Riesz space with 'T generated by the family of seminorms [pi}. is Fatou's full norm, then every p extension p. to L) is the Fatou full norm on L. Recall that a locally solid Riesz space (L, r) is called locally ordered complete if it has a basis of a neighborhood system of the origin. Let (L, -r) be a Hausdorff locally ordered complete Riesz space. L, -r) is also a locally ordered complete Riesz space.
The result will now follow immediately if we prove that L is a Dedekind-complete Riesz space.
