On the Closed Graph Theorem and the Open Mapping Theorem
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209 a Note on Closed Graph Theorems
Acta Math. Univ. Comenianae 209 Vol. LXXV, 2(2006), pp. 209–218 A NOTE ON CLOSED GRAPH THEOREMS F. GACH Abstract. We give a common generalisation of the closed graph theorems of De Wilde and of Popa. 1. Introduction In the theory of locally convex spaces, M. De Wilde’s notion of webs is the abstrac- tion of all that is essential in order to prove very general closed graph theorems. Here we concentrate on his theorem in [1] for linear mappings with bornologically closed graph that have an ultrabornological space as domain and a webbed lo- cally convex space as codomain. Henceforth, we shall refer to this theorem as ‘De Wilde’s closed graph theorem’. As far as the category of bounded linear mappings between separated convex bornological spaces is concerned, there exists a corresponding bornological notion of so-called nets that enabled N. Popa in [5] to prove a bornological version of the closed graph theorem which we name ‘Popa’s closed graph theorem’. Although M. De Wilde’s theorem in the locally convex and N. Popa’s theorem in the convex bornological setting are conceptually similar (see also [3]), they do not directly relate to each other. In this manuscript I present a bornological closed graph theorem that generalises the one of N. Popa and even implies the one of M. De Wilde. First of all, I give a suitable definition of bornological webs on separated convex bornological spaces with excellent stability properties and prove the aforementioned general bornological closed graph theorem. It turns out that there is a connection between topological webs in the sense of M. -
Uniform Boundedness Principle for Unbounded Operators
UNIFORM BOUNDEDNESS PRINCIPLE FOR UNBOUNDED OPERATORS C. GANESA MOORTHY and CT. RAMASAMY Abstract. A uniform boundedness principle for unbounded operators is derived. A particular case is: Suppose fTigi2I is a family of linear mappings of a Banach space X into a normed space Y such that fTix : i 2 Ig is bounded for each x 2 X; then there exists a dense subset A of the open unit ball in X such that fTix : i 2 I; x 2 Ag is bounded. A closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings as consequences of this principle. Some applications of this principle are also obtained. 1. Introduction There are many forms for uniform boundedness principle. There is no known evidence for this principle for unbounded operators which generalizes classical uniform boundedness principle for bounded operators. The second section presents a uniform boundedness principle for unbounded operators. An application to derive Hellinger-Toeplitz theorem is also obtained in this section. A JJ J I II closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings in the third section as consequences of this principle. Go back Let us assume the following: Every vector space X is over R or C. An α-seminorm (0 < α ≤ 1) is a mapping p: X ! [0; 1) such that p(x + y) ≤ p(x) + p(y), p(ax) ≤ jajαp(x) for all x; y 2 X Full Screen Close Received November 7, 2013. 2010 Mathematics Subject Classification. Primary 46A32, 47L60. Key words and phrases. -
Sequential Completeness of Inductive Limits
Internat. J. Math. & Math. Sci. Vol. 24, No. 6 (2000) 419–421 S0161171200003744 © Hindawi Publishing Corp. SEQUENTIAL COMPLETENESS OF INDUCTIVE LIMITS CLAUDIA GÓMEZ and JAN KUCERAˇ (Received 28 July 1999) Abstract. A regular inductive limit of sequentially complete spaces is sequentially com- plete. For the converse of this theorem we have a weaker result: if indEn is sequentially complete inductive limit, and each constituent space En is closed in indEn, then indEn is α-regular. Keywords and phrases. Regularity, α-regularity, sequential completeness of locally convex inductive limits. 2000 Mathematics Subject Classification. Primary 46A13; Secondary 46A30. 1. Introduction. In [2] Kuˇcera proves that an LF-space is regular if and only if it is sequentially complete. In [1] Bosch and Kuˇcera prove the equivalence of regularity and sequential completeness for bornivorously webbed spaces. It is natural to ask: can this result be extended to arbitrary sequentially complete spaces? We give a partial answer to this question in this paper. 2. Definitions. Throughout the paper {(En,τn)}n∈N is an increasing sequence of Hausdorff locally convex spaces with continuous identity maps id : En → En+1 and E = indEn its inductive limit. A topological vector space E is sequentially complete if every Cauchy sequence {xn}⊂E is convergent to an element x ∈ E. An inductive limit indEn is regular, resp. α-regular, if each of its bounded sets is bounded, resp. contained, in some constituent space En. 3. Main results Theorem 3.1. Let each space (En,τn) be sequentially complete and indEn regular. Then indEn is sequentially complete. Proof. Let {xk} be a Cauchy sequence in indEn. -
View of This, the Following Observation Should Be of Some Interest to Vector Space Pathologists
Can. J. Math., Vol. XXV, No. 3, 1973, pp. 511-524 INCLUSION THEOREMS FOR Z-SPACES G. BENNETT AND N. J. KALTON 1. Notation and preliminary ideas. A sequence space is a vector subspace of the space co of all real (or complex) sequences. A sequence space E with a locally convex topology r is called a K- space if the inclusion map E —* co is continuous, when co is endowed with the product topology (co = II^Li (R)*). A i£-space E with a Frechet (i.e., complete, metrizable and locally convex) topology is called an FK-space; if the topology is a Banach topology, then E is called a BK-space. The following familiar BK-spaces will be important in the sequel: ni, the space of all bounded sequences; c, the space of all convergent sequences; Co, the space of all null sequences; lp, 1 ^ p < °°, the space of all absolutely ^-summable sequences. We shall also consider the space <j> of all finite sequences and the linear span, Wo, of all sequences of zeros and ones; these provide examples of spaces having no FK-topology. A sequence space is solid (respectively monotone) provided that xy £ E whenever x 6 m (respectively m0) and y £ E. For x £ co, we denote by Pn(x) the sequence (xi, x2, . xn, 0, . .) ; if a i£-space (E, r) has the property that Pn(x) —» x in r for each x G E, then we say that (£, r) is an AK-space. We also write WE = \x Ç £: P„.(x) —>x weakly} and ^ = jx Ç E: Pn{x) —>x in r}. -
Some Simple Applications of the Closed Graph Theorem
SHORTER NOTES The purpose of this department is to publish very short papers of an unusually elegant and polished character, for which there is normally no other outlet. SOME SIMPLE APPLICATIONS OF THE CLOSED GRAPH THEOREM JESÚS GIL DE LAMADRID1 Our discussion is based on the following two simple lemmas. Our field of scalars can be either the reals or the complex numbers. Lemma 1. Let Ebea Banach space under a norm || || and Fa normed space under a norm || ||i. Suppose that || H2is a second norm of F with respect to which F is complete and such that || ||2^^|| \\ifor some k>0. Assume finally that T: E—+Fis a linear transformation which is bounded with respect toll II and II IK. Then T is bounded with respect to\\ \\ and Proof. The product topology oî EX F with respect to || || and ||i is weaker (coarser) than the product topology with respect to || and || 112.Since T is bounded with respect to j| || and || ||i, its graph G is (|| | , II ||i)-closed. Hence G is also (|| ||, || ||2)-closed. Therefore T is ( | [[, || ||2)-bounded by the closed graph theorem. Lemma 2. Assume that E is a Banach space under a norm || ||i and that Fis a vector subspace2 of E, which is a Banach space under a second norm || ||2=è&|| ||i, for some ft>0. Suppose that T: E—>E is a bounded operator with respect to \\ \\i and \\ ||i smcA that T(F)EF. Then the re- striction of T to F is bounded with respect to || H2and || ||2. Proof. -
Approximation Boundedness Surjectivity
APPROXIMATION BOUNDEDNESS SURJECTIVITY Olav Kristian Nygaard Dr. scient. thesis, University of Bergen, 2001 Approximation, Boundedness, Surjectivity Olav Kr. Nygaard, 2001 ISBN 82-92-160-08-6 Contents 1Theframework 9 1.1 Separability, bases and the approximation property ....... 13 1.2Thecompletenessassumption................... 16 1.3Theoryofclosed,convexsets................... 19 2 Factorization of weakly compact operators and the approximation property 25 2.1Introduction............................. 25 2.2 Criteria of the approximation property in terms of the Davis- Figiel-Johnson-Pe'lczy´nskifactorization.............. 27 2.3Uniformisometricfactorization.................. 33 2.4 The approximation property and ideals of finite rank operators 36 2.5 The compact approximation property and ideals of compact operators.............................. 39 2.6 From approximation properties to metric approximation prop- erties................................. 41 3 Boundedness and surjectivity 49 3.1Introduction............................. 49 3.2Somemorepreliminaries...................... 52 3.3 The boundedness property in normed spaces . ....... 57 3.4ThesurjectivitypropertyinBanachspaces........... 59 3.5 The Seever property and the Nikod´ymproperty......... 63 3.6 Some results on thickness in L(X, Y )∗ .............. 63 3.7Somequestionsandremarks.................... 65 4 Slices in the unit ball of a uniform algebra 69 4.1Introduction............................. 69 4.2Thesliceshavediameter2..................... 70 4.3Someremarks........................... -
Arxiv:2006.03419V2 [Math.GM] 3 Dec 2020 H Eain Ewe Opeeesadteeitneo fixe of [ Existence [7]
COMPLETENESS IN QUASI-PSEUDOMETRIC SPACES – A SURVEY S. COBZAS¸ Abstract. The aim of this paper is to discuss the relations between various notions of se- quential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. 17 (1967), 226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., 49 (1984), 36. A discussion on nets defined over ordered or pre-ordered directed sets is also included. Classification MSC 2020: 54E15 54E25 54E50 46S99 Key words: metric space, quasi-metric space, uniform space, quasi-uniform space, Cauchy sequence, Cauchy net, Cauchy filter, completeness 1. Introduction It is well known that completeness is an essential tool in the study of metric spaces, particularly for fixed points results in such spaces. The study of completeness in quasi-metric spaces is considerably more involved, due to the lack of symmetry of the distance – there are several notions of completeness all agreing with the usual one in the metric case (see [18] or [6]). Again these notions are essential in proving fixed point results in quasi-metric spaces as it is shown by some papers on this topic as, for instance, [4], [10], [16], [21] (see also the book [1]). A survey on the relations between completeness and the existence of fixed points in various circumstances is given in [7]. -
S-Barrelled Topological Vector Spaces*
Canad. Math. Bull. Vol. 21 (2), 1978 S-BARRELLED TOPOLOGICAL VECTOR SPACES* BY RAY F. SNIPES N. Bourbaki [1] was the first to introduce the class of locally convex topological vector spaces called "espaces tonnelés" or "barrelled spaces." These spaces have some of the important properties of Banach spaces and Fréchet spaces. Indeed, a generalized Banach-Steinhaus theorem is valid for them, although barrelled spaces are not necessarily metrizable. Extensive accounts of the properties of barrelled locally convex topological vector spaces are found in [5] and [8]. In the last few years, many new classes of locally convex spaces having various types of barrelledness properties have been defined (see [6] and [7]). In this paper, using simple notions of sequential convergence, we introduce several new classes of topological vector spaces which are similar to Bourbaki's barrelled spaces. The most important of these we call S-barrelled topological vector spaces. A topological vector space (X, 2P) is S-barrelled if every sequen tially closed, convex, balanced, and absorbing subset of X is a sequential neighborhood of the zero vector 0 in X Examples are given of spaces which are S-barrelled but not barrelled. Then some of the properties of S-barrelled spaces are given—including permanence properties, and a form of the Banach- Steinhaus theorem valid for them. S-barrelled spaces are useful in the study of sequentially continuous linear transformations and sequentially continuous bilinear functions. For the most part, we use the notations and definitions of [5]. Proofs which closely follow the standard patterns are omitted. 1. Definitions and Examples. -
Topologies on the Test Function Algebra Publications Du Département De Mathématiques De Lyon, 1975, Tome 12, Fascicule 1 , P
PUBLICATIONS DU DÉPARTEMENT DE MATHÉMATIQUES DE LYON G. LASSNER O∗-Topologies on the Test Function Algebra Publications du Département de Mathématiques de Lyon, 1975, tome 12, fascicule 1 , p. 25-38 <http://www.numdam.org/item?id=PDML_1975__12_1_25_0> © Université de Lyon, 1975, tous droits réservés. L’accès aux archives de la série « Publications du Département de mathématiques de Lyon » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Publications du Dlpartement de Math6matiques Lyon 1975 t.12.1 0*-TOPOLOGIES ON THE TEST FUNCTION ALGEBRA G. LASSNER 1. INTRODUCTION This paper deals with C*-like topologies on the test function algebra ?l » the tensor algebra over the Schwartz space [S . These topologies are connected with continuous representations of this algebra. If R is a *-algebra and 8s a pre-Hilbert space, then a representation a A(a) of R in if is a homomorphism of R into End £ , such that <$fA(a)lf>> « <A(a*)<frfiJ;> for all $ ^ e 3) . For a representation of a Banach *-algebra R all operators A(a) are automatically bounded and the representation is uniformly continuous. In fact, for 4>c® , f(a) = <<f>,A(a)<J» is a positive functional on R and therefore [ l] : ||A(aH|| = «t>,A(a*a)<j?> = f(a*a) * f<l)||a<a|| «N«H|2 l|a||2 , 25 O* -topologies on the test function algebra which implies j !A(a)| U||a|| . -
Continuity Properties and Sensitivity Analysis of Parameterized Fixed
Continuity properties and sensitivity analysis of parameterized fixed points and approximate fixed points Zachary Feinsteina Washington University in St. Louis August 2, 2016 Abstract In this paper we consider continuity of the set of fixed points and approximate fixed points for parameterized set-valued mappings. Continuity properties are provided for the fixed points of general multivalued mappings, with additional results shown for contraction mappings. Further analysis is provided for the continuity of the approxi- mate fixed points of set-valued functions. Additional results are provided on sensitivity of the fixed points via set-valued derivatives related to tangent cones. Key words: fixed point problems; approximate fixed point problems; set-valued continuity; data dependence; generalized differentiation MSC: 54H25; 54C60; 26E25; 47H04; 47H14 1 Introduction Fixed points are utilized in many applications. Often the parameters of such models are estimated from data. As such it is important to understand how the set of fixed points aZachary Feinstein, ESE, Washington University, St. Louis, MO 63130, USA, [email protected]. 1 changes with respect to the parameters. We motivate our general approach by consider applications from economics and finance. In particular, the solution set of a parameterized game (i.e., Nash equilibria) fall under this setting. Thus if the parameters of the individuals playing the game are not perfectly known, sensitivity analysis can be done in a general way even without unique equilibrium. For the author, the immediate motivation was from financial systemic risk models such as those in [10, 8, 4, 12] where methodology to estimate system parameters are studied in, e.g., [15]. -
On Nonbornological Barrelled Spaces Annales De L’Institut Fourier, Tome 22, No 2 (1972), P
ANNALES DE L’INSTITUT FOURIER MANUEL VALDIVIA On nonbornological barrelled spaces Annales de l’institut Fourier, tome 22, no 2 (1972), p. 27-30 <http://www.numdam.org/item?id=AIF_1972__22_2_27_0> © Annales de l’institut Fourier, 1972, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 22, 2 (1972), 27-30. ON NONBORNOLOGICAL BARRELLED SPACES 0 by Manuel VALDIVIA L. Nachbin [5] and T. Shirota [6], give an answer to a problem proposed by N. Bourbaki [1] and J. Dieudonne [2], giving an example of a barrelled space, which is not bornological. Posteriorly some examples of nonbornological barrelled spaces have been given, e.g. Y. Komura, [4], has constructed a Montel space which is not bornological. In this paper we prove that if E is the topological product of an uncountable family of barrelled spaces, of nonzero dimension, there exists an infinite number of barrelled subspaces of E, which are not bornological.We obtain also an analogous result replacing « barrelled » by « quasi-barrelled ». We use here nonzero vector spaces on the field K of real or complex number. The topologies on these spaces are sepa- rated. -
Metrizable Barrelled Spaces, by J . C. Ferrando, M. López Pellicer, And
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 34, Number 1, January 1997 S 0273-0979(97)00706-4 Metrizable barrelled spaces,byJ.C.Ferrando,M.L´opez Pellicer, and L. M. S´anchez Ruiz, Pitman Res. Notes in Math. Ser., vol. 332, Longman, 1995, 238 pp., $30.00, ISBN 0-582-28703-0 The book targeted by this review is the next monograph written by mathemati- cians from Valencia which deals with barrelledness in (metrizable) locally convex spaces (lcs) over the real or complex numbers and applications in measure theory. This carefully written and well-documented book complements excellent mono- graphs by Valdivia [21] and by P´erez Carreras, Bonet [12] in this area. Although for the reader some knowledge of locally convex spaces is requisite, the book under re- view is self-contained with all proofs included and may be of interest to researchers and graduate students interested in locally convex spaces, normed spaces and mea- sure theory. Most chapters contain some introductory facts (with proofs), which for the reader greatly reduces the need for original sources. Also a stimulating Notes and Remarks section ends each chapter of the book. The classical Baire category theorem says that if X is either a complete metric space or a locally compact Hausdorff space, then the intersection of countably many dense, open subsets of X is a dense subset of X. This theorem is a principal one in analysis, with applications to well-known theorems such as the Closed Graph Theorem, the Open Mapping Theorem and the Uniform Boundedness Theorem.