Semi-Indefinite-Inner-Product and Generalized Minkowski Spaces
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On Quasi Norm Attaining Operators Between Banach Spaces
ON QUASI NORM ATTAINING OPERATORS BETWEEN BANACH SPACES GEUNSU CHOI, YUN SUNG CHOI, MINGU JUNG, AND MIGUEL MART´IN Abstract. We provide a characterization of the Radon-Nikod´ymproperty in terms of the denseness of bounded linear operators which attain their norm in a weak sense, which complement the one given by Bourgain and Huff in the 1970's. To this end, we introduce the following notion: an operator T : X ÝÑ Y between the Banach spaces X and Y is quasi norm attaining if there is a sequence pxnq of norm one elements in X such that pT xnq converges to some u P Y with }u}“}T }. Norm attaining operators in the usual (or strong) sense (i.e. operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and also compact operators satisfy this definition. We prove that strong Radon-Nikod´ymoperators can be approximated by quasi norm attaining operators, a result which does not hold for norm attaining operators in the strong sense. This shows that this new notion of quasi norm attainment allows to characterize the Radon-Nikod´ymproperty in terms of denseness of quasi norm attaining operators for both domain and range spaces, completing thus a characterization by Bourgain and Huff in terms of norm attaining operators which is only valid for domain spaces and it is actually false for range spaces (due to a celebrated example by Gowers of 1990). A number of other related results are also included in the paper: we give some positive results on the denseness of norm attaining Lipschitz maps, norm attaining multilinear maps and norm attaining polynomials, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions to obtain that quasi norm attaining operators are actually norm attaining, study the relationship with the norm attainment of the adjoint operator and, finally, present some stability results. -
FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality). -
On Fixed Points of One Class of Operators
⥬ â¨ç÷ âã¤÷ù. .7, ü2 Matematychni Studii. V.7, No.2 517.988 ON FIXED POINTS OF ONE CLASS OF OPERATORS Yu.F. Korobe˘ınik Yu. Korobeinik. On xed points of one class of operators, Matematychni Studii, 7(1997) 187{192. Let T : Q ! Q be a selfmapping of some subset Q of a Banach space E. The operator T does not increase the distance between Q and the origin (the null element E) if 8x 2 Q kT xk ≤ kxk (such operator is called a NID operator). In this paper some sucient conditions for the existence of a non-trivial xed point of a NID operator are obtained. These conditions enable in turn to prove some new results on xed points of nonexpanding operators. Let Q be a subset of a linear normed space (E; k · k). An operator T : Q ! E is said to be nonexpanding or NE operator if 8x1; x2 2 Q kT x1 − T x2k ≤ kx1 − x2k: It is evident that each nonexpanding mapping of Q is continuous in Q. The problem of existence of xed points of nonexpanding operators was investi- gated in many papers (see [1]{[9]). The monograph [10] contains a comparatively detailed survey of essential results on xed points of NE operators obtained in the beginning of the seventies. In particular, it was proved that each NE operator in a closed bounded convex subset of a uniformly convex space has at least one xed point in Q (for the de nition of a uniformly convex space see [10], ch.2, x4 or [11], ch.V, x12). -
A Note on Bi-Contractive Projections on Spaces of Vector Valued Continuous
Concr. Oper. 2018; 5: 42–49 Concrete Operators Research Article Fernanda Botelho* and T.S.S.R.K. Rao A note on bi-contractive projections on spaces of vector valued continuous functions https://doi.org/10.1515/conop-2018-0005 Received October 10, 2018; accepted December 12, 2018. Abstract: This paper concerns the analysis of the structure of bi-contractive projections on spaces of vector valued continuous functions and presents results that extend the characterization of bi-contractive projec- tions given by the rst author. It also includes a partial generalization of these results to ane and vector valued continuous functions from a Choquet simplex into a Hilbert space. Keywords: Contractive projections, Bi-contractive projections, Vector measures, Spaces of vector valued functions MSC: 47B38, 46B04, 46E40 1 Introduction We denote by C(Ω, E) the space of all continuous functions dened on a compact Hausdor topological space Ω with values in a Banach space E, endowed with the norm YfY∞ = maxx∈Ω Yf(x)YE. The continuity of a function f ∈ C(Ω, E) is understood to be “in the strong sense", meaning that for every w0 ∈ Ω and > 0 there exists O, an open subset of Ω, containing w0 such that Yf (w) − f(w0)YE < , for every w ∈ O, see [9] or [16]. A contractive projection P ∶ C(Ω, E) → C(Ω, E) is an idempotent bounded linear operator of norm 1. ⊥ A contractive projection is called bi-contractive if its complementary projection P (= I − P) is contractive. In this paper, we extend the characterization given in [6], of bi-contractive projections on C(Ω, E), with Ω a compact metric space, E a Hilbert space that satisfy an additional support disjointness property. -
On the Continuity of Fourier Multipliers on the Homogeneous Sobolev Spaces
R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Krystian KAZANIECKI & Michał WOJCIECHOWSKI On the continuity of Fourier multipliers on the homogeneous Sobolev spaces . 1 d W1 (R ) Tome 66, no 3 (2016), p. 1247-1260. <http://aif.cedram.org/item?id=AIF_2016__66_3_1247_0> © Association des Annales de l’institut Fourier, 2016, Certains droits réservés. Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE. http://creativecommons.org/licenses/by-nd/3.0/fr/ L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 66, 3 (2016) 1247-1260 ON THE CONTINUITY OF FOURIER MULTIPLIERS . 1 d ON THE HOMOGENEOUS SOBOLEV SPACES W1 (R ) by Krystian KAZANIECKI & Michał WOJCIECHOWSKI (*) Abstract. — In this paper we prove that every Fourier multiplier on the ˙ 1 d homogeneous Sobolev space W1 (R ) is a continuous function. This theorem is a generalization of the result of A. Bonami and S. Poornima for Fourier multipliers, which are homogeneous functions of degree zero. Résumé. — Dans cet article, nous prouvons que chaque multiplicateur de Fou- ˙ 1 d rier sur l’espace homogène W1 (R ) de Sobolev est une fonction continue. Notre théorème est une généralisation du résultat de A. -
Weak Convergence of Asymptotically Homogeneous Functions of Measures
h’on,inec,r ~ndysrr, Theory. Methods & App/;cmonr. Vol. IS. No. I. pp. I-16. 1990. 0362-546X/90 13.M)+ .OO Printed in Great Britain. 0 1990 Pergamon Press plc WEAK CONVERGENCE OF ASYMPTOTICALLY HOMOGENEOUS FUNCTIONS OF MEASURES FRANGOISE DEMENGEL Laboratoire d’Analyse Numerique, Universite Paris-Sud, 91405 Orsay, France and JEFFREY RAUCH* University of Michigan, Ann Arbor, Michigan 48109, U.S.A. (Received 28 April 1989; received for publication 15 September 1989) Key words and phruses: Functions of measures, weak convergence, measures. INTRODUCTION FUNCTIONS of measures have already been introduced and studied by Goffman and Serrin [8], Reschetnyak [lo] and Demengel and Temam [5]: especially for the convex case, some lower continuity results for the weak tolopology (see [5]) permit us to clarify and solve, from a mathematical viewpoint, some mechanical problems, namely in the theory of elastic plastic materials. More recently, in order to study weak convergence of solutions of semilinear hyperbolic systems, which arise from mechanical fluids, we have been led to answer the following ques- tion: on what conditions on a sequence pn of bounded measures have we f(~,) --*f(p) where P is a bounded measure and f is any “function of a measure” (not necessarily convex)? We answer this question in Section 1 when the functions f are homogeneous, and in Section 2 for sublinear functions; the general case of asymptotically homogeneous function is then a direct consequence of the two previous cases! An extension to x dependent functions is described in Section 4. In the one dimensional case, we give in proposition 3.3 a criteria which is very useful in practice, namely for the weak continuity of solutions measures of hyperbolic semilinear systems with respect to weakly convergent Cauchy data. -
G = U Ytv2
PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 81, Number 1, January 1981 A CHARACTERIZATION OF MINIMAL HOMOGENEOUS BANACH SPACES HANS G. FEICHTINGER Abstract. Let G be a locally compact group. It is shown that for a homogeneous Banach space B on G satisfying a slight additional condition there exists a minimal space fimm in the family of all homogeneous Banach spaces which contain all elements of B with compact support. Two characterizations of Bm¡1¡aie given, the first one in terms of "atomic" representations. The equivalence of these two characterizations is derived by means of certain (bounded) partitions of unity which are of interest for themselves. Notations. In the sequel G denotes a locally compact group. \M\ denotes the (Haar) measure of a measurable subset M G G, or the cardinality of a finite set. ®(G) denotes the space of continuous, complex-valued functions on G with compact support (supp). For y G G the (teft) translation operator ly is given by Lyf(x) = f(y ~ xx). A translation invariant Banach space B is called a homogeneous Banach space (in the sense of Katznelson [9]) if it is continuously embedded into the topological vector space /^(G) of all locally integrable functions on G, satisfies \\Lyf\\B= 11/11,for all y G G, and lim^iy-- f\\B - 0 for all / G B (hence, as usual, two measurable functions coinciding l.a.e. are identified). If, furthermore, B is a dense subspace of LX(G) it is called a Segal algebra (in the sense of Reiter [15], [16]). Any homogeneous Banach space is a left L'((7)-Banach- module with respect to convolution, i.e./ G B, h G LX(G) implies h */ G B, and \\h */\\b < ll*Uill/IJi» m particular any Segal algebra is a Banach ideal of LX(G). -
Chebyshev Sets
CHEBYSHEV SETS JAMES FLETCHER and WARREN B. MOORS (October 9, 2014) Abstract A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this article we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the article we present a construction of a non-convex Chebyshev subset of an inner product space. 1. Introduction Given a normed linear space (X, ), we shall say that a subset of X is a ‘Chebyshev set’ if k·k every point outside of it admits a unique nearest point. In this article we will review some of the literature, and a little, a very little, of the history of the question of whether every Chebyshev set is convex (for a brief summary of the history of this problem, see [26, p.307]). The volume of research invested in this topic is vast and we will not even pretend to present, even a small amount, of what is known. Instead, we will be content to present some of the most basic results concerning Chebyshev sets (see Section 2) and then present what the authors believe are among some of the most interesting partial solutions to the following question, which from hence forth, will be referred to as the ‘Chebyshev set problem’: “Is every Chebyshev set in a Hilbert space convex?”. -
Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov–Galerkin, and Monotone Mixed Methods
Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov{Galerkin, and Monotone Mixed Methods I. Muga† · K.G. van der Zee‡ 5th June, 2018 Abstract This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residual minimization in dual norms, and its inex- act version using discrete dual norms. It is shown that this development, in the case of strictly-convex reflexive Banach spaces with strictly-convex dual, gives rise to a class of nonlinear Petrov{Galerkin methods and, equiv- alently, abstract mixed methods with monotone nonlinearity. Crucial in the formulation of these methods is the (nonlinear) bijective duality map. Under the Fortin condition, we prove discrete stability of the abstract inexact method, and subsequently carry out a complete error analysis. As part of our analysis, we prove new bounds for best-approximation projec- tors, which involve constants depending on the geometry of the underlying Banach space. The theory generalizes and extends the classical Petrov{ Galerkin method as well as existing residual-minimization approaches, such as the discontinuous Petrov{Galerkin method. Keywords Operators in Banach spaces · Residual minimization · Petrov{Galerkin discretization · Error analysis · Quasi-optimality · Duality mapping · Best approximation · Geometric constants Mathematics Subject Classification 41A65 · 65J05 · 46B20 · 65N12 · 65N15 arXiv:1511.04400v3 [math.NA] 29 Jul 2018 †Pontificia Univ. Cat´olicade Valpara´ıso,Instituto de Mathem´aticas,[email protected] ‡University of Nottingham, School of Mathematical Sciences, [email protected] 1 Contents 1 Introduction3 1.1 Petrov{Galerkin discretization and residual minimization . -
Math.FA] 21 May 2019 Now We Introduce the Notation and Necessary Preliminaries
STRONG SUBDIFFERENTIABILITY AND LOCAL BISHOP-PHELPS-BOLLOBAS´ PROPERTIES SHELDON DANTAS, SUN KWANG KIM, HAN JU LEE, AND MARTIN MAZZITELLI Abstract. It has been recently presented in [21] some local versions of the Bishop-Phelps-Bollob´as type property for operators. In the present article, we continue studying these properties for multilinear mappings. We show some differences between the local and uniform versions of the Bishop-Phelps- Bollob´astype results for multilinear mappings, and also provide some interesting examples which shows that this study is not just a mere generalization of the linear case. We study those properties for bilinear forms on `p × `q using the strong subdifferentiability of the norm of the Banach space `p⊗^ π`q. Moreover, we present necessary and sufficient conditions for the norm of a Banach space Y to be strongly N subdifferentiable through the study of these properties for bilinear mappings on `1 × Y . 1. Introduction In Banach space theory, it is well-known that the set of all norm attaining continuous linear functionals defined on a Banach space X is dense in its topological dual space X∗. This is the famous Bishop-Phelps theorem [9]. In 1970, this result was strengthened by Bollob´as,who proved a quantitative version in the following sense: if a norm-one linear functional x∗ almost attains its norm at some x, then, near to x∗ and x, there are, respectively, a new norm-one functional y∗ and a new point y such that y∗ attains its norm at y (see [10, Theorem 1]). Nowadays, this result is known as the Bishop-Phelps-Bollob´astheorem and it has been used as an important tool in the study of Banach spaces and operators. -
Arxiv:1310.3382V1 [Math.FA] 12 Oct 2013 Umn A,Bnc Space
SOME PROBLEMS IN FUNCTIONAL ANALYSIS INSPIRED BY HAHN BANACH TYPE THEOREMS M.A. SOFI1 ∗ Dedicated to Professor T. Ando with affection. Abstract. As a cornerstone of functional analysis, Hahn Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications of the Hahn Banach theorem which are less familiar to the mathematical community, apart from highlighting certain aspects of the Hahn Banach phenomena which have spurred intense research activity over the past few years, especially involving operator analogues and nonlinear variants of this theorem. For a discussion of a whole lot of issues related to the Hahn Banach theorem not treated in this paper, the best source is a famous survey paper by Narici and Beckenstein [31] which deals, among other things, with the different settings witnessing the validity of the Hahn Banach theorem. Contents 1. What is known to be folklore. 2. Unconventional applications of Hahn Banach theorem. 3. Some less known aspects of Hahn-Banach Theorem. 4. Hahn Banach-extension property as a Finite Dimensional property. 5. Hilbert spaces determined via Hahn Banach phenomena. arXiv:1310.3382v1 [math.FA] 12 Oct 2013 6. Intersection of balls and extendibility of maps. 1. What is known to be folklore Definition 1.1. Hahn Banach Theorem (HBT) (Classical, Real Case) Given a real normed linear space X, a subspace Y of X, a continuous linear Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. -
On Fermat-Torricelli Problem in Frechet Spaces on Fermat-Torricelli Problem in Frechet Spaces
On Fermat-Torricelli Problem in Frechet Spaces On Fermat-Torricelli Problem in Frechet Spaces I. A. Osinuga1, S. A. Ayinde2, J. A. Oguntuase3 and G. A. Adebayo4 1;3 Department of Mathematics, Federal University of Agriculture, Abeokuta, Ogun State, Nigeria 2 Basic Sciences Department, Babcock University, Ilishan-Remo, Ogun State, Nigeria. 4 Department of Physics, Federal University of Agriculture, Abeokuta, Ogun State, Nigeria. Correspondence to: S. A. Ayinde, Email: [email protected] Abstract: We study the Fermat-Torricelli problem (FTP) for Frechet space X, where X is considered as an inverse limit of projective system of Banach spaces. The FTP is defined by using fixed countable collec- tion of continuous seminorms that defines the topology of X as gauges. For a finite set A in X consisting of n distinct and fixed points, the set of minimizers for the sum of distances from the points in A to a variable point is considered. In particular, for the case of collinear points in X, we prove the existence of the set of minimizers for FTP in X and for the case of non collinear points, existence and uniqueness of the set of minimizers are shown for reflexive space X as a result of strict convexity of the space. Keywords: Reflexive Frechet spaces, Strict convexity, Fermat-Torricelli problem, Set of minimizers DOI: https://doi.org/10.3126/jnms.v3i2.33956 1 Introduction In 1643, Pierre de Fermat [17], proposed a problem to an Italian Physicist, Evangelista Torricelli. The problem was about finding the point the sum of whose distances from the vertices of a triangle is a minimum.