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Öğe All topologies come from a family of 0-1-valued quasi-metrics(Springer Singapore Pte Ltd, 2019) Ercan, Zafer; Vural, MehmetWe prove the statement in the title. This reproves that every topological space is induced by a quasi-uniformity.Öğe An answer to a question on the affine bijections on C(X, I)(Natl Inquiry Services Centre Pty Ltd, 2009) Ercan, Zafer; Önal, SüleymanA complete description of the bijective a. ne map on C(X, I) is given. This provides an answer to a question of [2] on the affine bijections on C(X, I).Öğe The Banach-Stone theorem revisited(Elsevier B.V. All, 2008) Ercan, Zafer; Önal, S.Let X and Y be compact Hausclorff spaces, and E and F be locally solid Riesz spaces. If pi : 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. Onal, 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. Onal, Banach-Stone theorem for Banach lattice valued continuous functions. Proc. Amer. Math. Soc. 135 (9) (2007) 2827-2829]. (C) 2008 Elsevier B.V. All rights reserved.Öğe A characterization of riesz n-morphisms and applications(Taylor & Francis Inc, 2008) Ercan, Zafer; Önal, SüleymanLet X-1 I X-2,..., X-n be realcompact spaces and Z he a topological space. Let pi : C(X-1)X C(X-2) X... X C(X-n)-> C(Z) be a Riesz n-morphism. We show that there exist functions sigma(i) : Z -> X-i (i = 1, 2,..., n) and w epsilon C(Z) such that pi(f(1), f(2),..., f(n)) = Wf(1) o sigma(1)f(2)o sigma(2)... fno sigma(n) and sigma(1), sigma(2),....,sigma(n) are continuous on {z : w(z)not equal 0}. This fact extends a result in Boulabiar (2002) and leads to one of the main results in Boulabiar (2004) with a more elementary proof.Öğe A characterization of u-uniformly completeness of Riesz spaces in terms of statistical U-uniformly pre-completeness(Walter de Gruyter GmbH, 2009) Ercan, ZaferIn this paper we introduce statistically u-uniformly convergent sequences in Riesz spaces (vector lattices) and then we give a characterization of u-uniformly completeness of Riesz spaces. © 2009 Warsaw University. All rights reserved.Öğe A CHARACTERIZATION OF u-UNIFORMLY COMPLETENESS OF RIESZ SPACES IN TERMS OF STATISTICAL u-UNIFORMLY PRE-COMPLETENESS(De Gruyter Open Ltd, 2009) Ercan, ZaferIn this paper we introduce statistically u-uniformly convergent sequences in Riesz spaces (vector lattices) and then we give a characterization of u-uniformly completeness of Riesz spaces.Öğe Characterizations of riesz spaces with b-property(Springer, 2009) Alpay, Şafak; Ercan, ZaferA Riesz space E is said to have b-property if each subset which is order bounded in E-similar to similar to is order bounded in E. The relationship between b-property and completeness, being a retract and the absolute weak topology vertical bar sigma vertical bar (E-similar to, E) is studied. Perfect Riesz spaces are characterized in terms of b-property. It is shown that b-property coincides with the Levi property in Dedekind complete Frechet lattices.Öğe A construction of hewitt realcompactification in terms of nets(Auburn University, 2014) Ercan, ZaferWe give a construction of Hewitt realcompactification of a completely regular Hausdorff space by using nets.Öğe Every norm is a restriction of an order-unit norm(2016) Çağlar, Mert; Ercan, ZaferWe point out the equivalence of the fact that every norm on a vector space is a restriction of an order-unit norm to that of Paulsen’s construction concerning generalization of operator systems.Öğe Fixing a gap in laterally closed lattice homomorphisms(Springer, 2022) Ercan, ZaferA gap in the proof of the main result of the paper A remark on the paper 'Laterally closed lattice homomorphisms' is fixed.Öğe Invariant subspaces for positive operators on locally convex solid Riesz spaces(Elsevier Science Bv, 2007) Çağlar, Mert; Ercan, ZaferWe prove that an x(0)-quasinilpotent semigroup S of continuous positive linear operators on a locally convex solid Riesz space X has a common invariant subspace. Using this, a result which implies the main theorem of Abramovich, Aliprantis and Burkinshaw [J. Funct. Anal. 115 (1993) 418-424] is also given.Öğe Mazur-Ulam theorem for riesz spaces(Walter De Gruyter Gmbh, 2010) Çelik, Cesim; Ercan, ZaferWe give a Mazur-Ulam type theorem for Riesz spaces. In particular, a generalization of the Mazur-Ulam theorem is given in terms of a lattice normed space.Öğe Memorandum on multiplicative bijections and order(Springer, 2009) Cabello Sanchez, Felix; Cabello Sanchez, Javier; Ercan, Zafer; Önal, SüleymanLet C(X, I) denote the semigroup of continuous functions from the topological space X to I = [0, 1], equipped with the pointwise multiplication. The paper studies semigroup homomorphisms C(Y, I) -> C(X, I), with emphasis on isomorphisms. The crucial observation is that, in this setting, homomorphisms preserve order, so isomorphisms preserve order in both directions and they are automatically lattice isomorphisms. Applications to uniformly continuous and Lipschitz functions on metric spaces are given. Sample result: if Y and X are complete metric spaces of finite diameter without isolated points, every multiplicative bijection T : Lip(Y, I) -> Lip(X, I) has the form Tf = f circle tau, where tau : X -> Y is a Lipschitz homeomorphism.Öğe A NEW PROOF OF THE SOBCZYK-HAMMER DECOMPOSITION THEOREM(Michigan State Univ Press, 2011) Ercan, ZaferIn this short note, we give a simple proof of the Sobczyk-Hammer Decomposition Theorem in terms of Dedekind complete Riesz spaces.Öğe A note on the main inclusion theorem of Luxemburg and Zaanen(2013) Ercan, ZaferWe give a more elementary proof of the main statement of the proof of the main result in [8]. Following the idea of [8], we re-prove a result of Luxemburg and Zaanen, that is, an Archimedean Riesz space is Dedekind complete if and only if it is uniformly complete and has the band projection propertyÖğe On Kakutani Krein and Maeda Ogasawara spaces(2014) Ercan, Zafer; Tan, Neşet Ozkan: Let E be an Archimedean Riesz space. It is shown that the Kakutani Krein space of the center of the Dedekind completion of E and the Maeda Ogasawara space of E are homeomorphic. By applying this, we can reprove a Banach Stone type theorem for C ∞(S) spaces, where S is a Stonean space.Öğe On Kakutani-Krein and Maeda-Ogasawara spaces(Scientific Technical Research Council Turkey-Tubitak, 2014) Ercan, Zafer; Tan, Neşet ÖzkanLet E be an Archimedean Riesz space. It is shown that the Kakutani-Krein space of the center of the Dedekind completion of E and the Maeda-Ogasawara space of E are homeomorphic. By applying this, we can reprove a Banach Stone type theorem for C-infinity (S) spaces, where S is a Stonean space.Öğe On the end of the cone metric space(Elsevier Science Bv, 2014) Ercan, ZaferStarting with the initial paper of Huang and Zhang [2] in 2007, more than six hundred papers dealing with cone metric spaces have been published so far. In this short note we present a different proof of the known fact that the notion of a cone metric space is not more general than that of a metric space. (C) 2014 Elsevier B.V. All rights reserved.Öğe ON THE LOS-MARCZEWSKI EXTENSION OF FINITELY ADDITIVE MEASURES(Michigan State Univ Press, 2011) Ercan, ZaferIn this short note, we give a relatively simple proof of the Los-Marczewski Extension of finitely additive measures. In particular, we extend the Los-Marczewski Extension to Dedekind complete Riesz-space-valued functions.Öğe Order-unit-metric Spaces(Springer, 2014) Çağlar, Mert; Ercan, ZaferWe study the concept of cone metric space in the context of ordered vector spaces by setting up a general and natural framework for it.
