Complete Vector Lattice Research Articles

Lattice-valued Borel measures. II

Let X be a completely regular T1 space, E a boundedly complete vector lattice, C(X) (Cb(X)) the space of all (all, bounded), real-valued continuous functions on X. In order convergence, we consider E-valued, order-bounded, -additive, -additive, and tight measures on X and prove some order-theoretic and topological properties of these measures. Also for an order-bounded, E-valued (for some special E) linear map on C(X), a measure representation result is proved. In case E n separates the points of E, an Alexanderov's type theorem is proved for a sequence of -additive measures.

Comparison of two types of order convergence with topological convergence in an ordered topological vector space

Birkhoff and Peressini proved that if ( X , T ) (X,\mathcal {T}) is a complete metrizable topological vector lattice, a sequence converges for the topology T \mathcal {T} iff the sequence relatively uniformly star converges. The above assumption of lattice structure is unnecessary. A necessary and sufficient condition for the conclusion is that the positive cone be closed, normal, and generating. If, moreover, the space ( X , T ) (X,\mathcal {T}) is locally convex, Namioka [11, Theorem 5.4] has shown that T \mathcal {T} coincides with the order bound topology T b {\mathcal {T}_b} and Gordon [4, Corollary, p. 423] (assuming lattice structure and local convexity) shows that metric convergence coincides with relative uniform star convergence. Omitting the assumptions of lattice structure and local convexity of ( X , T ) (X,\mathcal {T}) it is shown for the nonnecessarily local convex topology T ru {\mathcal {T}_{{\text {ru}}}} that T b ⊂ T ru = T {\mathcal {T}_{\text {b}}} \subset {\mathcal {T}_{{\text {ru}}}} = \mathcal {T} and T b = T ru = T {\mathcal {T}_{\text {b}}} = {\mathcal {T}_{{\text {ru}}}} = \mathcal {T} when ( X , T ) (X,\mathcal {T}) is locally convex.

A type of nearest point set in a complete 𝑙-group

A theorem by W. D. L. Appling (Riv. Mat. Univ. Parma, (3) 2 (1973), 251-276) demonstrates that a C-set is a nearest point set in ba ( S , Σ ) {\text {ba}}(S,\Sigma ) with respect to the variation norm. This paper demonstrates an analogous result for a generalized form of C-set in a complete l-group with distance with respect to the norm being replaced by a stronger property definable in a complete l-group (distance between elements x and y of a complete l-group G is taken to be | x − y | |x - y| ). The result is then shown to be a characterization of sets possessing the stronger property in the case of a complete vector lattice, but not a characterization in the case of a complete l-group.

The Jordan Decomposition of Vector-Valued Measures

This paper gives criteria for a vector-valued Jordan decomposition theorem to hold. In particular, suppose L is an order complete vector lattice and $\mathcal {A}$ is a Boolean algebra. Then an additive set function $\mu :\mathcal {A} \to L$ can be expressed as the difference of two positive additive measures if and only if $\mu (\mathcal {A})$ is order bounded. A sufficient condition for a countably additive set function with values in ${c_0}(\Gamma )$, for any set $\Gamma$, to be decomposed into difference of countably additive set functions is given; namely, the domain being the power set of some set.

Lattice-valued Borel measures

Let X be a completely regular T1 space, E a boundedly complete vector lattice, C(X) (Cb(X)) the space of all (all, bounded), real-valued continuous functions on X. In order convergence, we consider E-valued, order-bounded, -additive, -additive, and tight measures on X and prove some order-theoretic and topological properties of these measures. Also for an order-bounded, E-valued (for some special E) linear map on C(X), a measure representation result is proved. In case E n separates the points of E, an Alexanderov's type theorem is proved for a sequence of -additive measures.

The Jordan decomposition of vector-valued measures

This paper gives criteria for a vector-valued Jordan decomposition theorem to hold. In particular, suppose L is an order complete vector lattice and 6 is a Boolean algebra. Then an additive set function ,u: 6 -* L can be expressed as the difference of two positive additive measures if and only if ,u(d) is order bounded. A sufficient condition for a countably additive set function with values in c0(F), for any set F, to be decomposed into difference of countably additive set functions is given; namely, the domain being the power set of some set. We are concerned here with vector-valued additive set functions defined on some sort of Boolean algebra with vector-values in a (generally order complete) vector lattice. The purpose of this note is to expose conditions that insure such measures can be written as a difference of positive measures, i.e., conditions for a vector-valued Jordan decomposition theorem to hold. For this reason, a measure that can be expressed as the difference of two positive additive measures will be called decomposable. The decomposability of vector measures per se was first studied by C. E. Rickart in a 1943 Duke Mathematical Journal article where he established a Lebesgue decomposition theorem for the class of strongly bounded additive measures. This result was later re-established (although it was not realized at the time) by J. J. Uhl, Jr. [10] who also presented a Yosida-Hewitt decomposition theorem for strongly bounded measures. In [3], Diestel and Faires exhibited several decomposition theorems of the Jordan type, however, they did not give necessary and sufficient conditions for the decomposability of a vector measure. Our first result supplies these conditions, although, as is indicated, only one part of the proof is new. So, let E be a Boolean algebra, S. the vector lattice of simple functions over E endowed with the uniform norm, and L a vector lattice. If tt: C -> L is an additive set function, then the operator T : S, -> L associated with yt is defined by n n T(, aic) = ai fL(ai) Received by the editors January 24, 1975 and, in revised form, February 12, 1976. AMS (MOS) subject classifications (1970). Primary 46G10; Secondary 28A25. Copyright 6) 1977, American Mathematical Society

Über lösbarkeitseigenschaften homogener linearer ungleiehungen in halbgeordneten räumen

Properites of the solutions of dual homogeneous linear inequality systems in order complete topological vector lattices are investigated. Especially, a feneralization of the Tucker complementary slackness theorem is proved. Various examples illustrate the application of the obtained resu lts to infinte homogeneou linear inequality systems and to infinte linear programming problems.

A duality theorem for a convex programming problem in order complete vector lattices

The problem “minimize f(x) -g(x)” is studied, where f is a convex function and g a concave function of a real vector space X into an order complete vector lattice Y. We define Y-valued functions f”(T) and gc( T) on the real vector space of linear mappings of X into Y and associate with the minimization problem the maximization problem “maximize gC(T) fC(T).” For X = R” and Y = R Fenchel [2, 31 showed that these programs are dual to each other. Our main result will be that, iff g is bounded below, the maximization problem has an optimal solution and the extreme values of both objective functions are equal. As a direct consequence we get a condition, saying that a point x is an optimal solution of our minimization problem if and only if f and g have a common subgradient at x. This result extends a theorem by Valadier [9], concerning the subdifferentiability of a convex function with values in an order complete vector lattice. Furthermore, we apply our theorem to prove an extension of the KuhnTucker Theorem for order complete vector lattices. To this we generalize an idea from Rockafellar [7]. Our result is similar to one given by Ritter [6]. An example is given, showing that the hypothesis of order completeness is necessary for our theorems.

Eine Verallgemeinerung des lemmas Von NEYMAN-PEARSON

A theorem about the structure of optimal solutions of linear programming problems with a finite number of scalar restrictions and with upper bounded variables in order complete topological vector lattices is proved. This theorem is a generalization of the well-known NEYMAN-PEARSON-Lemma.

Konjugierte operatoren und subdifferentiale

In this paper we give separation theorems for convex sets of a product space. Separation is carried out by linear operators. With these theorems we prove several assertions on conjugate operators and subdifferentials of operators (which map a vector space into an order complete vector lattice), where we use the definition of conjugate operators as done by Zowe. Simultaneously we generalize some of his results to such operators. Moreover we prove that an order complete vector lattice is a vector lattice with certain separation properties.

A Banach lattice not weakly projectable

In [4] a concept of a weakly projectable vector lattice has been introduced. Stone vector lattices [3] and thus all special types of them, like Riesz [5], σ-complete and complete vector lattices are weakly projectable. Moreover C[0, 1] is weakly projectable but not Stone [4]. As we see the collection W of weakly projectable vector lattices is quite large. This explains to some extent the difficulty in producing examples of vector lattices which do not belong to W. In this note an example of a Banach lattice [1] which is not weakly projectable is described.

A note on the modulus of an order bounded linear operator between complex vector lattices

The vector space L b ( E, F) of all order bounded linear operators from the complex vector lattice E into the order complete complex vector lattice F is a complex vector lattice. The modulus of an order bounded linear operator T is defined by |T|= sup {Re(λT)|λϵC,|λ| = 1} Under additional conditions on E, some formulas are proved for | T|( x), where T ϵ L b ( E, F) and 0≦ x ϵ E. An example: If E is Archimedean and relatively uniformly complete, T ϵ L b ( E, F) and 0≦ x ϵ E, then | T|( x)=sup {| Ty| y ϵ E, | y| ≦ x}. These formulas were already known for σ-order complete complex vector lattices and for E= C( X), X compact Hausdorff.

A Note on Commutative l-Groups

Let G be a commutative lattice ordered group. Theorem 1 gives necessary and sufficient conditions under which a⊥ with a∈G is a maximal l-ideal. A wide family of, l-groups G having the property that the orthogonal complement of each atom is a maximal l-ideal is described. Conditionally σ-complete and hence conditionally complete vector lattices belong to the family.It follows immediately that if a is an atom in a conditionally complete vector lattice then a⊥ is a maximal vector lattice ideal. This theorem has been proved in [7] by Yamamuro. Theorem 2 generalizes another result contained in [7]. Namely we prove that if M is a closed maximal l-ideal of an archimedean l-group G then there exists an atom a ∈ G such that M = a⊥.

An existence theorem for strong liftings

Let (2, CL) be a couple consisting of a locally compact Hausdorff space 2 and a positive Radon measure p on 2. By Pm@, CL) we denote the algebra of bounded P-measurable functions on 2 in the sense of Bourbaki [I], by L”(Z, cl) their classes modulo locally negligible functions, and by N” the essential supremums norm on SP’(Z, p) and L”(Z, p); L”(Z, p) is a Banach algebra and a complete vector lattice [2, Chap. I].

Multiplication in Vector Lattices

B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)).

Spectral decomposition of quasi-Montel spaces

In [8] the author showed that Montel spaces have the property that all regular Borel spectral measures with values in their continuous-linear-transformation algebras are necessarily purely atomic. The purpose of this note is to make the observation that by virtue of a theorem of Bartle, Dunford and Schwartz [1] and Grothendieck [3], this property is shared by a significantly larger class of locally convex spaces, namely the quasi-Montel spaces of K. Kera [5]. This class includes the classical Banach space 1 and its subspaces and the gestufte Raiume of Kothe and their subspaces [6]. The result given in this note also has consequences, which we shall mention briefly, in the study of the singular operators of Kantorovitz [4]. Let E [Z] be a locally convex topological vector space, which will be assumed to be boundedly complete; recall that a spectral measure triple in S(E) is a triple (X, 25, ,u) where X is a set, 25 is a a-algebra of subsets of X, and ,u: -->.C(E) is an S(E)-valued set function which is countably additive in the weak operator topology and for which ,u(X) = 1 (the identity transformation) and for any 5, E-C2, Mu(6'e) =-(8),u(E). (X, (25, ,u) is said to be equicontinuous if the values of ,u on (25 are, and to be Baire or Borel if X is a compact Hausdorff space and (5 = e or Q8, its a-algebras of Baire or Borel sets respectively. A Borel spectral measure triple (X, 03, ,u) is said to be regular if (Mu(.)x, x') is a regular Borel measure for each xCE and x'CE' (cf. [8, Proposition 3.18]). A point atom of a Borel measure is a point t CX with .( {I }) #0. For x CE, the cyclic subspace and real cyclic subspace generated by x, denoted by 91Q(x) and J)R(X) respectively, are the smallest closed subspace and closed real subspace of E respectively which contain { u(8)xl}es. [8, Proposition 3.15] shows that both 9NR(X) and 9(x) are complete locally convex spaces when E is boundedly complete in Z, [8, Proposition 3.13 et seq.] that )R(X) is a complete vector lattice when ordered by taking its positive cone to be the closed convex cone generated by { u(8)x1}sE, and also that 91Q(X) = =9RR(X) ED i fR(X) X It is easy to see that only small modifications of the proof of [8, Theorem 4.1] suffice to yield a proof of the slightly stronger-looking

The Hahn-Banach theorem for finite dimensional spaces

1. Introduction. An immediate generalization of the Hahn-Banach Theorem on the dominated extension of linear functionals is obtained by replacing the real number field by a boundedly complete vector lattice as the range space of the function. The question arises: Do boundedly complete vector lattices characterize those range spaces which permit dominated extensions? It is proved in [1], [4] that an ordered linear space is a boundedly complete vector lattice if and only if both the positive wedge is lineally closed and the Hahn-Banach Theorem holds with the space taken as the range of the extensions. Thus, the above question has an affirmative answer if the positive wedge must be closed in a space for which the Hahn-Banach Theorem is valid. This paper shows that at least for finite dimensional ordered linear spaces this is indeed the case. An example is presented in [4] of a two dimensional ordered linear space whose positive wedge is not lineally closed and it is erroneously asserted that this space permits Hahn-Banach type extensions. The error in the argument is that the extensions described may not be well defined. ?2 introduces preliminary definitions and theorems. In ?3, finite dimensional boundedly complete vector lattices are characterized by the existence of a partly positive basis for the positive wedge. This characterization is a slight generalization of a theorem of Yudin [5] or Nagy [3]. In ?4 it is proved that if a finite dimensional ordered linear space permits Hahn-Banach type extensions, then the closure of the positive wedge determines an ordered linear space which is a boundedly complete vector lattice. This restricts the class of positive wedges which must be considered in ascertaining the class which admits dominated extensions. Examples are given in ?5 which are used in ?6 to complete the proof that a finite dimensional ordered linear space admits dominated extensions if and only if the space is a boundedly complete vector lattice.

On Köthe spaces

1. Kothe and Toeplitz in [7] and later Kothe in [8] and [9] began the study of certain pairs of subspaces of the space of all real sequences. Their theory has been generalized by Dieudonne [4], M. J. Cooper [3], and Lorentz and Wertheim [11]. Dieudonne, in his work considers pairs (A, A*) of subspaces of the space of locally integrable functions on a locally compact (-compact Hausdorff space E with a Radon measure ,u. The spaces (A, A*), which are locally convex topologically complete vector lattices, are called K6the spaces; well-known examples of these are the pairs (Lp, Lq). We present two results which contribute to the theory as developed by Dieudonne; however, since the topology of the underlying topological space E plays no role in these results, they are presented in a purely measure theoretic setting. This necessitates presenting some of the general theory of K6the spaces. The first of the two results generalizes the classical theorem: if Lrc LS and r # s, then L' is of the first category in Ls; the second result proves the equivalence of four statements; one of which is: A* can be identified with the topological dual of A.

Эргодическая теорема в линейных полуупорядоченных пространствах с абстрактной нормой

Эргодическая теорема в линейных полуупорядоченных пространствах с абстрактной нормой

Duality in vector lattices

Introduction. We shall prove an abstract theorem on the duality of vector lattices. It is powerful and general enough to yield quickly the duality of both the classical examples of concrete vector lattices as well as various abstract analogs of these lattices. The theorem treats the duality of L1 and Lo, so as to exhibit the relation of their duality to the duality of the Lp spaces for 1 xa >p= xlaa =x, then limf(xa) =f(x). This implies that f is bounded in the sense that the set {f(x): 0 <x <y } is bounded for each y. It follows that f and the zero functional have a supremum and infimum relative to the lattice of all bounded linear functionals [1, p. 245]. The concepts of completeness and o-continuity arise naturally in a duality theory for this reason: the set B of all bounded linear functionals on a vector lattice is easily seen to be a complete vector lattice, a rising (directed upward) set of linear functionals converging pointwise to its supremum; the same is true for the subset of B consisting of the functions continuous in a topology T such that if U is a neighborhood of 0, there is a neighborhood V of 0 with { I xl xC V} C U. In addition, if x is an element in the original lattice, the functional on B induced by x is o-continuous. We now give an important definition, which is followed by the main theorem. Recall that a nondegenerate bilinear functional on the cartesian product of two vector spaces L and M, is one such that the maps x +x* and y--+y* defined below are 1-1 onto their range. DEFINITION. A pair of complete vector lattices, (L, M) is said to be paired with respect to the nondegenerate bilinear functional (x, y) on LXM provided the functionals y*: x-+(x, y) and x*: y-+(x, y) are ocontinuous for x CL, y E M, and each vector lattice is a sublattice of the lattice of all bounded linear functionals on the other, i.e., the functional