Non-Archimedean Closed Graph Theorems
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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. -
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]. -
1.2.3 Banach–Steinhaus Theorem Another Fundamental Theorem Of
16 1 Basic Facts from Functional Analysis and Banach Lattices 1.2.3 Banach{Steinhaus Theorem Another fundamental theorem of functional analysis is the Banach{Steinhaus theorem, or the Uniform Boundedness Principle. It is based on a fundamental topological results known as the Baire Category Theorem. Theorem 1.13. Let X be a complete metric space and let fXngn≥1 be a sequence of closed subsets in X. If Int Xn = ; for any n ≥ 1, then 1 S Int Xn = ;. Equivalently, taking complements, we can state that a count- n=1 able intersection of open dense sets is dense. Remark 1.14. Baire's theorem is often used in the following equivalent form: if X is a complete metric space and fXngn≥1 is a countable family of closed 1 S sets such that Xn = X, then IntXn 6= ; at least for one n. n=1 Chaotic dynamical systems We assume that X is a complete metric space, called the state space. In gen- eral, a dynamical system on X is just a family of states (x(t))t2T parametrized by some parameter t (time). Two main types of dynamical systems occur in applications: those for which the time variable is discrete (like the observation times) and those for which it is continuous. Theories for discrete and continuous dynamical systems are to some extent parallel. In what follows mainly we will be concerned with continuous dynam- ical systems. Also, to fix attention we shall discuss only systems defined for t ≥ 0, that are sometimes called semidynamical systems. Thus by a contin- uous dynamical system we will understand a family of functions (operators) (x(t; ·))t≥0 such that for each t, x(t; ·): X ! X is a continuous function, for each x0 the function t ! x(t; x0) is continuous with x(0; x0) = x0. -
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]. -
1 the Principal of Uniform Boundedness
THE PRINCIPLE OF UNIFORM BOUNDEDNESS 1 The Principal of Uniform Boundedness Many of the most important theorems in analysis assert that pointwise hy- potheses imply uniform conclusions. Perhaps the simplest example is the theorem that a continuous function on a compact set is uniformly contin- uous. The main theorem in this section concerns a family of bounded lin- ear operators, and asserts that the family is uniformly bounded (and hence equicontinuous) if it is pointwise bounded. We begin by defining these terms precisely. Definition 1.1 Let A be a family of linear operators from a normed space X to a normed space Y . We say that A is pointwise bounded if supA2AfkAxkg < 1 for every x 2 X. We say A is uniformly bounded if supA2AfkAkg < 1. It is possible for a single linear operator to be pointwise bounded without being bounded (Exercise), so the hypothesis in the next theorem that each individual operator is bounded is essential. Theorem 1.2 (The Principle of Uniform Boundedness) Let A ⊆ L(X; Y ) be a family of bounded linear operators from a Banach space X to a normed space Y . Then A is uniformly bounded if and only if it is pointwise bounded. Proof We will assume that A is pointwise bounded but not uniformly bounded, and obtain a contradiction. For each x 2 X, define M(x) := supA2AfkAxkg; our assumption is that M(x) < 1 for every x. Observe that if A is not uniformly bounded, then for any pair of positive numbers and C there must exist some A 2 A with kAk > C/, and hence some x 2 X with kxk = but kAxk > C. -
Arxiv:1808.01914V1 [Math.FA] 3 Aug 2018 Eirus Elpsdes Oal Ovxspaces
STATIONARY DENSE OPERATORS IN SEQUENTIALLY COMPLETE LOCALLY CONVEX SPACES DANIEL VELINOV, MARKO KOSTIC,´ AND STEVAN PILIPOVIC´ Abstract. The purpose of this paper is to investigate the stationary dense operators and their connection to distribution semigroups and abstract Cauchy problem in sequentially complete spaces. 1. Introduction and Preliminaries In [15] and [12] are given definitions of a distribution semigroups with densely and non densely defined generators. Stationary dense operators in a Banach space E and their connections to densely defined distribution semigroups are introduced and analyzed in [11]. Our main aim in this paper is to consider such operators when E is sequentially complete locally convex space. When we are dealing with a semigroup in a locally convex space, in order to have the existence of the resolvent of an infinitesimal generator A, we suppose that the semigroup is equicontinuous. Moreover for A, we are led to study the behavior of A∞ in D∞(A). If A is stationary dense, the information lost in passing from A in E to A∞ in D∞(A), can be retrieved. Let us emphasize that the interest for the stationarity of A comes out from the solvability of the Cauchy problem u′ = Au, n u(0) = u0, for every u ∈ D(A ) for some n. This is noted in Proposition 2.3. In Theorem 2.12 we have a partial answer to the converse question: Whether the solvability for every x ∈ D(An) implies n-stationarity. Also, we extend the results of [2], [11], [18] and give new examples of stationary dense operators on sequentially complete locally convex spaces. -
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. -
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E extracta mathematicae Vol. 32, N´um.1, 1 { 24 (2017) A Note on Some Isomorphic Properties in Projective Tensor Products Ioana Ghenciu Mathematics Department, University of Wisconsin-River Falls, Wisconsin, 54022, USA [email protected] Presented by Jes´usM. F. Castillo Received May 23, 2016 Abstract: A Banach space X is sequentially Right (resp. weak sequentially Right) if every Right subset of X∗ is relatively weakly compact (resp. weakly precompact). A Banach space X has the L-limited (resp. the wL-limited) property if every L-limited subset of X∗ is relatively weakly compact (resp. weakly precompact). We study Banach spaces with the weak sequentially Right and the wL-limited properties. We investigate whether the projective tensor product of two Banach spaces X and Y has the sequentially Right property when X and Y have the respective property. Key words: R-sets, L-limited sets, sequentially Right spaces, L-limited property. AMS Subject Class. (2010): 46B20, 46B25, 46B28. 1. Introduction A bounded subset A of a Banach space X is called a Dunford-Pettis (DP) ∗ ∗ (resp. limited) subset of X if every weakly null (resp. w -null) sequence (xn) in X∗ tends to 0 uniformly on A; i.e., ( ) lim supfjx∗ (x)j : x 2 Ag = 0: n n A sequence (xn) is DP (resp. limited) if the set fxn : n 2 Ng is DP (resp. limited). A subset S of X is said to be weakly precompact provided that every sequence from S has a weakly Cauchy subsequence. Every DP (resp. limited) set is weakly precompact [37, p.