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1.
We show that a sequentially (τ)-complete topological vector lattice Xτ is isomorphic to some L1(μ), if and only if the positive cone can be written as X+ = +B for some convex, (τ)-bounded, and (τ)-closed set B X+ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y+ = + (for u >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.  相似文献   

2.
In classical topology it is proved, nonconstructively, that for a topological space X, every bounded Riesz map ϕ in C(X) is of the form for a point xX. In this paper our main objective is to give the pointfree version of this result. In fact, we constructively represent each real Riesz map on a compact frame M by prime elements. Received March 23, 2004; accepted in final form May 14, 2005.  相似文献   

3.
Let X and Y be compact Hausdorff spaces, and E and F be locally solid Riesz spaces. If π:C(X,E)→C(Y,F) is a 1-biseparating Riesz isomorphism then X and Y are homeomorphic, and E and F are Riesz isomorphic. This generalizes the main results of [Z. Ercan, S. Önal, Banach-Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (9) (2007) 2827-2829] and [X. Miao, C. Xinhe, H. Jiling, Banach-Stone theorems and Riesz algebras, J. Math. Anal. Appl. 313 (1) (2006) 177-183], and answers a conjecture in [Z. Ercan, S. Önal, Banach-Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (9) (2007) 2827-2829].  相似文献   

4.
Suppose that X and Y are Banach spaces complemented in each other with supplemented subspaces A and B. In 1996, W. T. Gowers solved the Schroeder–Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain some suitable conditions involving the spaces A and B to yield that X is isomorphic to Y or to provide that at least X m is isomorphic to Yn for some m, n ∈ IN*. So we get some decomposition methods in Banach spaces via supplemented subspaces resembling Pełczyński’s decomposition methods. In order to do this, we introduce several notions of Schroeder–Bernstein Quadruples acting on the spaces X, Y, A and B. Thus, we characterize them by using some Banach spaces recently constructed. Received: October 4, 2005.  相似文献   

5.
《Quaestiones Mathematicae》2013,36(4):403-418
Abstract

We find a necessary and sufficient condition on a Riesz space E such that a sub-Boolean algebra of components of a positive element generates the Boolean algebra of all components of the element. This condition yields a dual characterization of principal A-modules in the sense of D. Vuza. When applied to l-algebras, one finds improvements of results of J. Synnatzschke.  相似文献   

6.
《Quaestiones Mathematicae》2013,36(3):307-321
ABSTRACT

We show that the functional calculus defined on the class of Dedekind σ-complete Riesz spaces can be extended to the class of uniformly complete Archimedean Riesz spaces without representing in the process the spaces involved by spaces of functions. As a consequence some results in the theory of Riesz spaces which were proved previously by representation techniques, can now be proved in an intrinsic way.  相似文献   

7.
We study the spectrum of forcing notions between the iterations of σ-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of α-proper forcings for indecomposable countable ordinals α, the Axiom A forcings and forcings completely embeddable into an iteration of a σ-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner?s Axiom A. This resolves a conjecture of Baumgartner from the 1980s.  相似文献   

8.
Duals Invert     
Monoidal objects (or pseudomonoids) in monoidal bicategories share many of the properties of the paradigmatic example: monoidal categories. The existence of (say, left) duals in a monoidal category leads to a dualization operation which was abstracted to the context of monoidal objects by Day et al. (Appl Categ Struct 11:229–260, 2003). We define a relative version of this called exact pairing for two arrows in a monoidal bicategory; when one of the arrows is an identity, the other is a dualization. In this context we supplement results of Day et al. (Appl Categ Struct 11:229–260, 2003) (and even correct one of them) and only assume the existence of biduals in the bicategory where necessary. We also abstract recent work of Day and Pastro (New York J Math 14:733–742, 2008) on Frobenius monoidal functors to the monoidal bicategory context. Our work began by focusing on the invertibility of components at dual objects of monoidal natural transformations between Frobenius monoidal functors. As an application of the abstraction, we recover a theorem of Walters and Wood (Theory Appl Categ 3:25–47, 2008) asserting that, for objects A and X in a cartesian bicategory , if A is Frobenius then the category Map(X,A) of left adjoint arrows is a groupoid. Also, the characterization in Walters and Wood (Theory Appl Categ 3:25–47, 2008) of left adjoint arrows between Frobenius objects of a cartesian bicategory is put into our current setting. In the same spirit, we show that when a monoidal object admits a dualization, its lax centre coincides with the centre defined in Street (Theory Appl Categ 13:184–190, 2004). Finally we look at the relationship between lax duals for objects and adjoints for arrows in a monoidal bicategory.  相似文献   

9.
《Quaestiones Mathematicae》2013,36(3):283-297
Abstract

We discuss the notion of equimeasurability in the general setting of Riesz spaces and obtain a characterization for (Carleman) abstract kernel operators in terms of equimea=surable sets.  相似文献   

10.
We give a short and direct proof for the computation of the Szlenk index of the C(K) spaces, when K is a countable compact space and determine their Lavrientiev indices. We also compute the Szlenk index of certain C(α) spaces, where α is an uncountable ordinal. Finally, we show that if the Szlenk index of a Banach space is ω (first infinite ordinal), then its weak*-dentability index is at most ω2 and that this estimate is optimal. The first author was supported by the grants: Institutional Research Plan AV0Z10190503, A100190502, GA ČR 201/04/0090.  相似文献   

11.
LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH −1 are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. Research supported by the Fundació Caixa Castelló, MI/25.043/92  相似文献   

12.
An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL. A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism. These properties were introduced by Schmidt in [5]. In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM. An analogous theorem is derived for Riesz spaces with extension property (E2).  相似文献   

13.
We show that two versions of a first countable topological space which are equivalent in ZFC set theory split in the absence of the Axiom of Choice AC. This answers in the negative a related question from Gutierres “What is a first countable space?”.  相似文献   

14.
A maximal antichain A of poset P splits if and only if there is a set BA such that for each pP either bp for some bB or pc for some cA\B. The poset P is cut-free if and only if there are no x < y < z in P such that [x,z]P = [x,y]P ∪ [y,z]P . By [1] every maximal antichain in a finite cut-free poset splits. Although this statement for infinite posets fails (see [2])) we prove here that if a maximal antichain in a cut-free poset “resembles” to a finite set then it splits. We also show that a version of this theorem is just equivalent to Axiom of Choice. We also investigate possible strengthening of the statements that “A does not split” and we could find a maximal strengthening. * This work was supported, in part, by Hungarian NSF, under contract Nos. T37846, T34702, T37758, AT 048 826, NK 62321. The second author was also supported by Bolyai Grant.  相似文献   

15.
system of simple types  , which uses the intuitionistic propositional calculus, with the only connective →. It is very important, because the well known Curry-Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property: every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard [4], under the name of system F, still with the normalization property. More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming languages. It is interesting to notice that, as early as 1972, Clint and Hoare [1] had made an analogous remark for the law of excluded middle and controlled jump instructions in imperative languages. There are now many type systems which are based on classical logic; among the best known are the system LC of J.-Y. Girard [5] and the λμ-calculus of M. Parigot [11]. We shall use below a system closely related to the latter, called the λ c -calculus [8, 9]. Both systems use classical second order logic and have the normalization property. In the sequel, we shall extend the λ c -calculus to the Zermelo-Fr?nkel set theory. The main problem is due to the axiom of extensionality. To overcome this difficulty, we first give the axioms of ZF in a suitable (equivalent) form, which we call ZF ɛ . Received: 6 September 1999 / Published online: 25 January 2001  相似文献   

16.
We prove weighted L p -inequalities for multi-parameter Riesz type potentials, strong fractional maximal operators and their dyadic counterparts. Our proofs avoid the Good-λ inequalities used earlier in the R m -case and are based on our integrated multi-parameter summation by parts lemma, that might be of independent interest.  相似文献   

17.
A question of Foreman and Magidor asks if it is consistent for every sequence of stationary subsets of the ns for 1n< to be mutually stationary. We get a positive answer to this question in the context of the negation of the Axiom of Choice. We also indicate how a positive answer to a generalized version of this question in a choiceless context may be obtained.The author wishes to thank James Cummings for helpful correspondence on the subject matter of this paper. The author also wishes to thank the referee and Andreas Blass, the corresponding editor, for helpful comments and suggestions that have been incorporated into this version of the paper. 03E35, 03E55 Supercompact cardinal – Indestructibility – Almost huge cardinal – Mutual stationarity – Symmetric inner modelRevised version: 6 June 2004  相似文献   

18.
We show that in the dual of Weak L1 the subspace of all rearrangement invariant continuous linear functionals is lattice isometric to a space L1(μ) and is the linear hull of the maximal elements of the dual unit ball. We also show that the dual of Weak L1 contains a norm closed weak* dense ideal which is lattice isometric to an 1-sum of spaces of type C(K). Helmut H. Schaefer in memoriam  相似文献   

19.
We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces LXp, where X is a Banach space and 1≤ p<∞, and extend the result to vector-valued Banach function spaces EX, where E is a Banach function space with order continuous norm. The author is supported by the ‘VIDI subsidie’ 639.032.201 in the ‘Vernieuwingsimpuls’ programme of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281.  相似文献   

20.
Let X and Y be Banach spaces such that each of them is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we give suitable conditions on finite sums of X and Y to yield that Xm is isomorphic to Yn for some In other words, we obtain some extensions of the well-known Pełczyński decomposition method in Banach spaces. In order to do this, we introduce the notion of Nearly Schroeder-Bernstein Quadruples for Banach spaces and pose a Conjecture to characterise them. Received: 5 January 2005  相似文献   

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