RACSAM. A note on FréchetUrysohn locally convex spaces. 1 Introduction. Jerzy Ka kol and M. López Pellicer


 Rose Perry
 1 years ago
 Views:
Transcription
1 RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 101 (2), 2007, pp Análisis Matemático / Mathematical Analysis A note on FréchetUrysohn locally convex spaces Jerzy Ka ol and M. López Pellicer Abstract. Recently Cascales, Ka ol and Saxon showed that in a large class of locally convex spaces (so called class G) every FréchetUrysohn space is metrizable. Since there exist (under Martin s axiom) nonmetrizable separable FréchetUrysohn spaces C p(x) and only metrizable spaces C p(x) belong to class G, we study another sufficient conditions for FréchetUrysohn locally convex spaces to be metrizable. Una nota sobre espacios de FréchetUrysohn localmente convexos Resumen. Recientemente Cascales, Ka ol y Saxon han probado que en una amplia clase de espacios localmente convexos (llamada clase G) los espacios con la propiedad de FréchetUrysohn son metrizables. Si se admite el axioma de Martin existen espacios C p(x) separables que tienen la propiedad de FréchetUrysohn y que no son metrizables. La metrizabilidad de los espacios C p(x) que pertenecen a la clase G ha motivado el que se estudie en este artículo condiciones suficientes para que los espacios localmente convexos con la propiedad de FréchetUrysohn sean metrizables. 1 Introduction One of the interesting and difficult problems (Malyhin 1978) concerning FréchetUrysohn groups ass if every separable FréchetUrysohn topological group is metrizable [18]; see also [20] and [21] for some counterexamples under various additional settheoretic assumptions. The same question can be formulated in the class of locally convex spaces (lcs). Under Martin s axiom (MA) there exist nonmetrizable analytic (hence separable) FréchetUrysohn spaces C p (X). On the other hand, the Borel Conjecture implies that separable and FréchetUrysohn spaces C p (X) are metrizable. In fact there exist many important classes of lcs for which the FréchetUrysohn property implies metrizablity. We showed in [3, Theorem 2] that (LM) spaces, (DF )spaces (in fact all spaces in class G) are metrizable if and only if they are FréchetUrysohn. In [22, Theorem 5. 7] Webb proved that only finitedimensional Montel (DF )spaces enjoy the Fréchet Urysohn property. We extend this fact by noticing that every FréchetUrysohn hemicompact topological group is a Polish space. The aim of the rest part of the paper is to characterize metrizability of Fréchet Urysohn lcs in terms of certain resolutions. First we prove that for a lcs X its strong dual F is metrizable if and only if F is FréchetUrysohn and X has a bornivorous bounded resolution. This applies to observe that the space of distributions D (Ω) and the space A(Ω) of real analytic functions on an open set Ω R N are not FréchetUrysohn (although they have countable tightness by [3, Corollary 2. 4]). Nevertheless, there exist FréchetUrysohn nonmetrizable lcs which admit a bornivorous bounded resolution, see Example 1. We show however that a FrćhetUrysohn lcs is metrizable if and only if it admits a superresolution. Presentado por Darío Maravall Casesnoves. Recibido: 3 de septiembre de Aceptado: 10 de octubre de Palabras clave / Keywords: Kanalytic space, analytic space, FréchetUrysohn space, topological group. Mathematics Subject Classifications: 46A30 c 2007 Real Academia de Ciencias, España. 127
2 J. Ka ol and M. López Pellicer 2 Notations, Definitions and Elementary Facts A Hausdorff topological space (space) X is said to have a compact resolution if X is covered by an ordered family {A α : α N N }, that is, such that A α A β for α β. Any Kanalytic space has a compact resolution but the converse implication fails in general [19]. For a lcs X a resolution {A α : α N N } is called bounded if each set A α is bounded in X. If additionally every bounded set in X is absorbed by some K α, the resolution {A α : α N N } is called bornivorous. For α = (n ) N N put C n1n 2...n := {Kβ : β = (m l ) l, m j = n j, j = 1,..., }. Clearly K α C n1,...,n for each N. A lcs X is said to have a superresolution if for every finite tuple (n 1,... n p ) of positive integers and every bounded set Q in X there exists α = (m ) N N such that m j = n j for 1 j p and K α absorbs Q. This implies that for any finite tuple (n 1,... n p ) the sequence (C n1,...,n p,n) n is bornivorous in X. Clearly any metrizable lcs X admits a superresolution: For a countable basis (U ) of neighbourhoods of zero in X and α = (n ) N N set K α := n U. Then {K α : α N N } is as required. Moreover, (DF )spaces, regular (LM)spaces admit a bornivorous bounded resolution. Indeed, let X be an (LM)space and let (X n ) n be an increasing sequence of metrizable lcs whose inductive limit is X, see [15] for details. For each n N let (U n) be a countable basis of absolutely convex neihbourhoods of zero in X n such that U n U n+1 and 2U+1 n U n for each N. For each α = (n ) N N set K α := n U n1. Then {K α : α N N } is a bounded resolution. In fact, {K α : α N N } covers X and the set n U n1 is bounded in X n1, hence in the limit space X. If additionally X is regular, i.e. for every bounded set B in X there exists m 1 N such that B is contained and bounded in X m1, then for each N there exists n N such that B n U m1. This yields a sequence α = (m ) N N such that B K α. Any lcs admitting a fundamental sequence (B n ) n of bounded sets has a superresolution; put K α := K n1 for α = (n ) N N. A space X is FréchetUrysohn if for each set A in X and any x A there exists a sequence of elements of A converging to x. A lcs X is barrelled (quasibarrelled) if every closed absolutely convex absorbing (bornivorous) subset of X is a neighbourhood of zero. A lcs X is called Bairelie [17] (bbairelie [16]) if for every increasing (and bornivorous) sequence (A n ) n of absolutely convex closed subsets of E there exists n N such that A n is a neighbourhood of zero in E. Every metrizable (metrizable and barrelled) lcs is bbairelie (Bairelie) and every barrelled bbairelie space is Bairelie. Every FréchetUrysohn lcs is both bbairelie and bornological, [9]. By C p (X) and C c (X) we denote the spaces of realvalued continuous functions on a Tychonov space endowed with the topology of pointwise convergence and the compactopen topology, respectively. 3 Results and Remars Note that for an uncountable compact scattered space X the space C p (X) is FréchetUrysohn [1, Theorem III. 1. 2] nonmetrizable and admits a bounded resolution. (**) Is a FréchetUrysohn lcs metrizable if it admits a strongly bounded resolution? Clearly every lcs with a fundamental sequence of bounded sets generates a bornivorous bounded resolution. In [3] we proved that every FréchetUrysohn lcs in class G is metrizable, so every FéchetUrysohn (DF )space is normable. Nevertheless, we have the following Example 1 Let X be a Σproduct of R I, with uncountable I, formed by all sequences of countable support. Then X contains a FréchetUrysohn nonmetrizable vector subspace having a bornivorous bounded resolution. PROOF. Clearly X is a FréchetUrysohn space, see [13]. Let G be the linear span of the compact set { B := [ 1, 1] I. Set E := F G and denote B F by C. For α = (n ) set K α = n 1 C. Note that Kα : α N N} is a strongly bounded resolution in E. First observe that the family { K α : α N N} is ordered, covers E and each K α is a bounded set in E. Next, let P be a bounded set in E and let A be its closed absolutely convex cover in E. Then A is bounded in R I, hence relatively compact in R I, and 128
3 A note on FréchetUrysohn locally convex spaces since every cluster point of a countable set in X has countable support, A is countably compact in X. But since A is closed in E, we conclude that A is a countably compact subset of E. It is easy to see that A is a Banach dis, i.e. its linear span E A endowed with Minowsi functional norm is a Banach space. Since {nc E A : n N} is a sequence of closed absolutely convex sets in E A covering E A and E A is a Baire space, there is m N such that A mc. So if β = (n ) N N verifies that n 1 = m, then A K β, so that P K β. Therefore the family {K α : α N N } is a strongly bounded resolution in E as stated and E is nonmetrizable since C is not metrizable. In general we note the following Proposition 1 For a lcs X its strong dual (X, β(x, X)) is metrizable if and only if it is FréchetUrysohn and X admits a bornivorous bounded resolution. PROOF. If (X, β(x, X)) is metrizable, then it is FréchetUrysohn and X admits a bornivorous bounded resolution. Now assume that X admits a bornivorous bounded resolution {K α : α N N }. Then the polars Kα of K α in the topological dual X of X form a base of neighbourhoods of zero for the strong topology β(x, X). But the polars Kα in X compose a resolution consisting of equicontinuous sets covering X. This shows that the space (X, β(x, X)) belongs to class G (in sense of Cascales and Orihuela of [2]). If (X, β(x, X)) is FréchetUrysohn, then [3, Theorem 2] yields the metrizability of (X, β(x, X)). Corollary 1 The spaces D (Ω) of distributions and A(Ω) of real analytic functions for an open set Ω R N are not FréchretUrysohn. PROOF. Since D (Ω) is nonmetrizable (quasibarrelled) and is the strong dual of a complete (LF )space D(Ω) of the test functions, we apply Proposition 1. The same argument can be used to the space A(Ω) via [5, Theorem 1. 7 and Proposition 1. 7]. Although (**) has a negative answer we note the following result for a large class of lcs. We need the following somewhat technical Proposition 2 A bbairelie space X is metrizable if and only if X admits a superresolution PROOF. Let {K α : α N N } be a superresolution for X. We may assume that all sets K α are absolutely convex. Then the sets C n1n 2...n (defined above) are also absolutely convex. First observe that for every α = (n ) N N and every neighbourhood of zero U in X there exists N such that C n1,n 2,...,n 2 U. Indeed, otherwise there exists a neighbourhood of zero U in X such that for every N there exists x C n1,n 2,...,n such that x / 2 U. Since x C n1,n 2,...,n for every N, there exists β = (m n) n N N such that x K β, n j = m j, j = 1, 2,...,. Set a n = max { m n : N } for n N and γ = (a n ) n. Since β γ for every N, then K β K γ, so x K γ for all N. Therefore 2 x 0 which provides a contradiction. The claim is proved. Let Y be the completion of X. It is clear that Y is Bairelie. Next, we show that there exists α = (n ) N N such that C n1,n 2,...,n is a neighbourhood of zero in X for each N. Assume that does not exist n 1 N such that C n1 is a neighbourhood of zero. Since (by assumption) the sequence (nc n ) n is bornivorous in X and X is quasibarrelled, we apply [15, ] to deduce that Y = X = n nc n (1 + ɛ) n nc n. But Y is a Bairelie space, so there exists n 1 N such that C n1 is a neighbourhood of zero in Y. Assume that for a finite tuple (n 1,... n p ) of positive integers the set C n1,...,n is a neighbourhood of zero for each 1 p. Since, by assumption, the sequence (nc n1,...,n p,n) n is bornivorous in X, and consequently X = n nc n 1,...,n p,n, we apply the same argument as above to get an integer n p+1 N such that C n1,...,n p,n p+1 is a neighbourhood of zero, which completes the inductive step. This fact combined with the claim provides a countable basis (2 C n1,...,n ) of neighbourhoods of zero, so X is metrizable. 129
4 J. Ka ol and M. López Pellicer The wea topology of any infinitedimensional normed space is nonmetrizable nonbornological but admits a superresolution. Since FréchetUrysohn lcs and spaces C p (X) are bbairelie, Proposition 2 applies to get the following Theorem 1 A FréchetUrysohn lcs is metrizable if and only if it admits a superresolution. C p (X) is metrizable if and only if C p (X) admits a superresolution. As we have already mentioned a FréchetUrysohn lcs in class G is metrizable. But only metrizable spaces C p (X) belong to class G, see [3]. On the other hand, (this fact might be already nown) under (MA)+ (CH) there exist nonmetrizable analytic FréchetUrysohn spaces C p (X). Indeed, by [1, Theorem II.3.2] the space C p (X) is FréchetUrysohn if and only if X has the γproperty, i.e. if for any open cover R of X such that any finite subset of X is contained in a member of R, there exists an infinite subfamily R of R such that any element of X lies in all but finitely many members of R. Gerlits and Nagy [7] showed that under (MA) every subset of reals of cardinality smaller than the continuum has the γproperty. Hence under MA+ (CH) there are uncountable γsubsets Y of reals, see also [8]. Thus for such Y the space C p (Y ) is nonmetrizable separable and FréchetUrysohn. A set of reals A is said to have strong measure zero if for any sequence (t n ) n of positive reals there exists a sequence of intervals (I n ) n covering A with I n < t n for each n N. The Borel Conjecture states that every strong measure zero set is countable; Laver [11] proved that it is relatively consistent with ZFC that the Borel conjecture is true. Assuming the Borel Conjecture, every FréchetUrysohn separable space C p (X) is metrizable. Indeed, since C p (X) is separable and FréchetUrysohn, X admits a weaer separable metric topology ξ and X has the γproperty. Set Y := (X, ξ). Since Y has the γproperty, then C p (Y ) is FréchetUrysohn [7]. By [12, Proposition 4 and Theorem 1] the space Y is zerodimensional. Since Y is metrizable and separable, it is homeomorphic to a subset Z with the γproperty of the Cantor set. On the other hand, GerlitsNagy [7] proved that γsets have strong measure zero. By the Borel Conjecture any γset in the reals is countable. Hence Z is countable. Thus X is countable and C p (X) is metrizable. Since FréchetUrysohn lcs are bbairelie, then every FréchetUrysohn lcs with a fundamental sequence of bounded sets must be a (DF )space. But FréhctUrysohn (DF )spaces are metrizable, [3]. For topological groups the situation is even more striing. The paper [21, Example 1.2] provides nonmetrizable FréchetUrysohn σcompact topological groups but for FréchetUrysohn hemicompact groups X the situation is different. Next proposition extends Webb s theorem [22, Theorem 5.7], who proved that only finitedimensional Montel (DF )spaces enjoy the FréchetUrysohn property. Proposition 3 Every FréchetUrysohn hemicompact topological group X is a locally compact Polish space. PROOF. Let (K n ) n be an increasing sequence of compact sets covering X such that every compact set in X is contained in some K m. Observe first that X is locally compact: It is enough to show that there exists n N such that K n is a neighborhood of the unit of X. Let F be a base of neighborhoods of the unit of X and assume that no K n contains a member of F. For every U F and n N choose x U,n U \ K n, and for each n N let A n = {x U,n : U F}. Since 0 A n for every n N, there exists a sequence (U m,n ) m in F such that x m,n 0, m, where x m,n := x Um,n,n. By [14, Theorem 4] there exists a sequence (n ) of distinct numbers in N and a sequence (m ) in N such that x m,n 0. Since {x m,n : N} is contained in some K p but x m,n / K n, N, we get a contradiction. Hence X is a locally compact FréchetUrysohn group. Since every locally compact FréchetUrysohn topological group is metrizable, see e.g. [10, Theorem 2], we conclude that X is analytic. But any analytic Baire topological group is a Polish space [4, Theorem 5.4], and we reach the conclusion. Acnowledgement. This research was supported for authors by the project MTM of the Spanish Ministry of Education and Science, cofinancied by the European Community (Feder funds). The first named author was also supported by the Grant MNiSW Nr. N N and Ayudas para Estancias en la U. P. V. de Investigadores de prestigio (2007). 130
5 A note on FréchetUrysohn locally convex spaces References [1] Arhangel sii, A. V., (1992). Topological function spaces, Math. and its Appl., Kluwer. [2] Cascales, B. and Orihuela, J., (1987). On Compactness in Locally Convex Spaces, Math. Z., 195, [3] Cascales, B., Ka ol, J. and Saxon, S. A., (2003). Metrizability vs. FréchetUrysohn property, Proc. Amer. Math. Soc., 131, [4] Christensen, J. P. R., (1974). Topology and Borel Structure, North Holland, Amsterdam, Math. Studies 10. [5] Domansi, P. and Vogt, D., (2001). Linear topological properties of the space of analytic functions on the real line, in Recent Progress in Functional Analysis, K. D. Bierstedt et al., eds., NorthHolland Math. Studies 189, [6] Fabian, M., Habala, P., Háje, P., Montesinos, V., Pelant, J. and Zizler, V., (2001). Functional Analysis and InfiniteDimensional Geometry, Canad. Math. Soc., Springer [7] Gerlits, J. and Nagy, Z. S., (1982). Some properties of C(X) I, Top. Appl., 14, [8] Galvin, F. and Miller, A. W., (1984). γsets and other singular sets of real numbers, Top. Appl., 17, [9] Ka ol, J. and Saxon, S., (2003). The FréchetUrysohn property, (LM)spaces and the strongest locally convex topology, Proc. Roy. Irish Acad., 103, 1 8. [10] Ka ol, J., LopezPellicer, M., Peinador, E. and Tarieladze, V., (2007). Lindelöf spaces C(X) over topological groups, Forum Math., DOI: /FORUM.2007.XXX, (to appear). [11] Laver, R.,(1976). On the consistence of Borel s conjecture, Acta Math., 137, [12] McCoy, R. A., (1980). space function spaces, Intern. J. Math., 3, [13] Noble, N., (1970). The continuity of functions on cartesian products, Trans. Math. Soc., 149, [14] Nyios, P. J., (1981). Metrizability and FréchetUrysohn property in topological groups, Proc. Amer. Math. Soc., 83, [15] Pérez Carreras, P. and Bonet, J., (1987). Barrelled Locally Convex Spaces, NorthHolland Math. Studies 131, Notas de Mat., 113, NorthHolland, Amsterdam. [16] Ruess, W., (1976). Generalized inductive limit topologies and barrelledness properties, Pacific J. of Math., 63, [17] Saxon, A., (1972). Nuclear and product spaces, Bairelie spaces, and the strongest locally convex topology, Math. Ann., 197, [18] Shahmatov, D., (2002). Convergence in the presence of algebraic structure, Recent Progress in Topology II, Amsterdam. [19] Talagrand, M., (1979). Espaces de Banach faiblement Kanalytiques, Ann. of Math., 110, [20] Todorčević, S. and Uzcátegui, C., (2005). Analytic spaces, Top. Appl., , [21] Todorčević, S. and Moore, J. T., (2006). The metrization problem for Fréchet groups, Preprint. [22] Webb, J. H.,(1968). Sequential convergence in locally convex spaces, Proc. Cambridge Phil. Soc., 64, Jerzy Ka ol M. López Pellicer Faculty of Mathematics and Informatics Depto. de Matemática Aplicada and IMPA A. Miciewicz University, Universidad Politécnica de Valencia Poznań, Poland E Valencia, Spain 131
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationINTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES
INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the
More informationA CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT
Bol. Soc. Mat. Mexicana (3) Vol. 12, 2006 A CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT F. MONTALVO, A. A. PULGARÍN AND B. REQUEJO Abstract. We present some internal conditions on a locally
More information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More informationON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction
ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationA Simple Observation Concerning Contraction Mappings
Revista Colombiana de Matemáticas Volumen 46202)2, páginas 229233 A Simple Observation Concerning Contraction Mappings Una simple observación acerca de las contracciones German LozadaCruz a Universidade
More informationChapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces
Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationMinimal R 1, minimal regular and minimal presober topologies
Revista Notas de Matemática Vol.5(1), No. 275, 2009, pp.7384 http://www.saber.ula.ve/notasdematematica/ Comisión de Publicaciones Departamento de Matemáticas Facultad de Ciencias Universidad de Los Andes
More informationThe MarkovZariski topology of an infinite group
Mimar Sinan Güzel Sanatlar Üniversitesi Istanbul January 23, 2014 joint work with Daniele Toller and Dmitri Shakhmatov 1. Markov s problem 1 and 2 2. The three topologies on an infinite group 3. Problem
More informationTHE REGULAR OPEN CONTINUOUS IMAGES OF COMPLETE METRIC SPACES
PACIFIC JOURNAL OF MATHEMATICS Vol 23 No 3, 1967 THE REGULAR OPEN CONTINUOUS IMAGES OF COMPLETE METRIC SPACES HOWARD H WICKE This article characterizes the regular T o open continuous images of complete
More informationThree observations regarding Schatten p classes
Three observations regarding Schatten p classes Gideon Schechtman Abstract The paper contains three results, the common feature of which is that they deal with the Schatten p class. The first is a presentation
More information1 Introduction, Notation and Statement of the main results
XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matemática Aplicada Sevilla, 2428 septiembre 2007 (pp. 1 6) Skewproduct maps with base having closed set of periodic points. Juan
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationCONVERGENCE OF SEQUENCES OF ITERATES OF RANDOMVALUED VECTOR FUNCTIONS
C O L L O Q U I U M M A T H E M A T I C U M VOL. 97 2003 NO. 1 CONVERGENCE OF SEQUENCES OF ITERATES OF RANDOMVALUED VECTOR FUNCTIONS BY RAFAŁ KAPICA (Katowice) Abstract. Given a probability space (Ω,
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationMultiplicative Relaxation with respect to Thompson s Metric
Revista Colombiana de Matemáticas Volumen 48(2042, páginas 227 Multiplicative Relaxation with respect to Thompson s Metric Relajamiento multiplicativo con respecto a la métrica de Thompson Gerd Herzog
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationMikołaj Krupski. Regularity properties of measures on compact spaces. Instytut Matematyczny PAN. Praca semestralna nr 1 (semestr zimowy 2010/11)
Mikołaj Krupski Instytut Matematyczny PAN Regularity properties of measures on compact spaces Praca semestralna nr 1 (semestr zimowy 2010/11) Opiekun pracy: Grzegorz Plebanek Regularity properties of measures
More informationAnalytic cohomology groups in top degrees of Zariski open sets in P n
Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationarxiv:math/0510680v3 [math.gn] 31 Oct 2010
arxiv:math/0510680v3 [math.gn] 31 Oct 2010 MENGER S COVERING PROPERTY AND GROUPWISE DENSITY BOAZ TSABAN AND LYUBOMYR ZDOMSKYY Abstract. We establish a surprising connection between Menger s classical covering
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationThe fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit
The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),
More informationSOME USES OF SET THEORY IN ALGEBRA. Stanford Logic Seminar February 10, 2009
SOME USES OF SET THEORY IN ALGEBRA Stanford Logic Seminar February 10, 2009 Plan I. The Whitehead Problem early history II. Compactness and Incompactness III. Deconstruction P. Eklof and A. Mekler, Almost
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationIrreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients
DOI: 10.2478/auom20140007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca
More informationSOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction
italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University
More informationOn the existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems
RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 97 (3), 2003, pp. 461 466 Matemática Aplicada / Applied Mathematics Comunicación Preliminar / Preliminary Communication On the existence of multiple principal
More informationPoint Set Topology. A. Topological Spaces and Continuous Maps
Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationOn the representability of the biuniform matroid
On the representability of the biuniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every biuniform matroid is representable over all sufficiently large
More informationSETTHEORETIC CONSTRUCTIONS OF TWOPOINT SETS
SETTHEORETIC CONSTRUCTIONS OF TWOPOINT SETS BEN CHAD AND ROBIN KNIGHT AND ROLF SUABEDISSEN Abstract. A twopoint set is a subset of the plane which meets every line in exactly two points. By working
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationFinite dimensional topological vector spaces
Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationMIKAEL LINDSTRÖM LIST OF PUBLICATIONS 10.4.2014. Articles in international peerreviewed journals
MIKAEL LINDSTRÖM LIST OF PUBLICATIONS 10.4.2014 Articles in international peerreviewed journals 2. Mikael Lindström: On Schwartz convergence vector spaces, Math. Nachr. 117, 3749, 1984. 3. Mikael Lindström:
More informationFinite Sets. Theorem 5.1. Two nonempty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we
More informationTHE UNIQUENESS OF HAAR MEASURE AND SET THEORY
C O L L O Q U I U M M A T H E M A T I C U M VOL. 74 1997 NO. 1 THE UNIQUENESS OF HAAR MEASURE AND SET THEORY BY PIOTR Z A K R Z E W S K I (WARSZAWA) Let G be a group of homeomorphisms of a nondiscrete,
More informationREAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE
REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).
More informationLEVERRIERFADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS
This is a reprint from Revista de la Academia Colombiana de Ciencias Vol 8 (106 (004, 3947 LEVERRIERFADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS by J Hernández, F Marcellán & C Rodríguez LEVERRIERFADEEV
More informationA simple application of the implicit function theorem
Boletín de la Asociación Matemática Venezolana, Vol. XIX, No. 1 (2012) 71 DIVULGACIÓN MATEMÁTICA A simple application of the implicit function theorem Germán LozadaCruz Abstract. In this note we show
More informationQuasi Contraction and Fixed Points
Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa00168, 6 Pages doi:10.5899/2012/jnaa00168 Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA Email: topolog@auburn.edu ISSN: 01464124
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More information1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1
Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between
More informationFALSE ALARMS IN FAULTTOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem
Opuscula Mathematica Vol. 32 No. 4 2012 http://dx.doi.org/10.7494/opmath.2012.32.4.751 FALSE ALARMS IN FAULTTOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of faulttolerant
More informationTree sums and maximal connected Ispaces
Tree sums and maximal connected Ispaces Adam Bartoš drekin@gmail.com Faculty of Mathematics and Physics Charles University in Prague Twelfth Symposium on General Topology Prague, July 2016 Maximal and
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationPrime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.
More informationThe sum of digits of polynomial values in arithmetic progressions
The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France Email: stoll@iml.univmrs.fr
More informationMath212a1010 Lebesgue measure.
Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationFINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS
FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure
More informationON A GLOBALIZATION PROPERTY
APPLICATIONES MATHEMATICAE 22,1 (1993), pp. 69 73 S. ROLEWICZ (Warszawa) ON A GLOBALIZATION PROPERTY Abstract. Let (X, τ) be a topological space. Let Φ be a class of realvalued functions defined on X.
More informationTwo classes of ternary codes and their weight distributions
Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationCURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationThe Stekloff Problem for Rotationally Invariant Metrics on the Ball
Revista Colombiana de Matemáticas Volumen 47(2032, páginas 890 The Steklo Problem or Rotationally Invariant Metrics on the Ball El problema de Steklo para métricas rotacionalmente invariantes en la bola
More informationFIXED POINT SETS OF FIBERPRESERVING MAPS
FIXED POINT SETS OF FIBERPRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 email: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More informationThe BanachTarski Paradox
University of Oslo MAT2 Project The BanachTarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the BanachTarski paradox states that for any ball in R, it is possible
More informationPowers of Two in Generalized Fibonacci Sequences
Revista Colombiana de Matemáticas Volumen 462012)1, páginas 6779 Powers of Two in Generalized Fibonacci Sequences Potencias de dos en sucesiones generalizadas de Fibonacci Jhon J. Bravo 1,a,B, Florian
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. FiniteDimensional
More informationON THE COEFFICIENTS OF THE LINKING POLYNOMIAL
ADSS, Volume 3, Number 3, 2013, Pages 4556 2013 Aditi International ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL KOKO KALAMBAY KAYIBI Abstract Let i j T( M; = tijx y be the Tutte polynomial of the matroid
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More informationEXTENSIONS OF MAPS IN SPACES WITH PERIODIC HOMEOMORPHISMS
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 4, July 1972 EXTENSIONS OF MAPS IN SPACES WITH PERIODIC HOMEOMORPHISMS BY JAN W. JAWOROWSKI Communicated by Victor Klee, November 29, 1971
More informationSome other convexvalued selection theorems 147 (2) Every lower semicontinuous mapping F : X! IR such that for every x 2 X, F (x) is either convex and
PART B: RESULTS x1. CHARACTERIZATION OF NORMALITY TYPE PROPERTIES 1. Some other convexvalued selection theorems In this section all multivalued mappings are assumed to have convex values in some Banach
More informationStationary random graphs on Z with prescribed iid degrees and finite mean connections
Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the nonnegative
More informationON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE
Illinois Journal of Mathematics Volume 46, Number 1, Spring 2002, Pages 145 153 S 00192082 ON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE J. HUISMAN Abstract. Let C be a
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationLecture Notes on Measure Theory and Functional Analysis
Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents
More informationCHAPTER 1 BASIC TOPOLOGY
CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationIntroducing Functions
Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1
More informationON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. GacovskaBarandovska
Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. GacovskaBarandovska Smirnov (1949) derived four
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationOn end degrees and infinite cycles in locally finite graphs
On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel
More informationThis chapter is all about cardinality of sets. At first this looks like a
CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },
More informationRate of growth of Dfrequently hypercyclic functions
Rate of growth of Dfrequently hypercyclic functions A. Bonilla Departamento de Análisis Matemático Universidad de La Laguna Hypercyclic Definition A (linear and continuous) operator T in a topological
More informationRecursion Theory in Set Theory
Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied
More informationAn algorithmic classification of open surfaces
An algorithmic classification of open surfaces Sylvain Maillot January 8, 2013 Abstract We propose a formulation for the homeomorphism problem for open ndimensional manifolds and use the Kerekjarto classification
More informationMetric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of
More informationA new continuous dependence result for impulsive retarded functional differential equations
CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas
More information