Approximation Boundedness Surjectivity
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Multivariable Approximation by Convolutional Kernel Networks
ITAT 2016 Proceedings, CEUR Workshop Proceedings Vol. 1649, pp. 118–122 http://ceur-ws.org/Vol-1649, Series ISSN 1613-0073, c 2016 V. K˚urková Multivariable Approximation by Convolutional Kernel Networks Veraˇ K˚urková Institute of Computer Science, Academy of Sciences of the Czech, [email protected], WWW home page: http://www.cs.cas.cz/ vera ∼ Abstract: Computational units induced by convolutional widths parameters) benefit from geometrical properties of kernels together with biologically inspired perceptrons be- reproducing kernel Hilbert spaces (RKHS) generated by long to the most widespread types of units used in neu- these kernels. These properties allow an extension of rocomputing. Radial convolutional kernels with vary- the maximal margin classification from finite dimensional ing widths form RBF (radial-basis-function) networks and spaces also to sets of data which are not linearly separable these kernels with fixed widths are used in the SVM (sup- by embedding them into infinite dimensional spaces [2]. port vector machine) algorithm. We investigate suitabil- Moreover, symmetric positive semidefinite kernels gener- ity of various convolutional kernel units for function ap- ate stabilizers in the form of norms on RKHSs suitable for proximation. We show that properties of Fourier trans- modeling generalization in terms of regularization [6] and forms of convolutional kernels determine whether sets of enable characterizations of theoretically optimal solutions input-output functions of networks with kernel units are of learning tasks [3, 19, 11]. large enough to be universal approximators. We com- Arguments proving the universal approximation prop- pare these properties with conditions guaranteeing positive erty of RBF networks using sequences of scaled kernels semidefinitness of convolutional kernels. -
GALERKIN APPROXIMATION in BANACH and HILBERT SPACES Wolfgang Arendt, Isabelle Chalendar, Robert Eymard
GALERKIN APPROXIMATION IN BANACH AND HILBERT SPACES Wolfgang Arendt, Isabelle Chalendar, Robert Eymard To cite this version: Wolfgang Arendt, Isabelle Chalendar, Robert Eymard. GALERKIN APPROXIMATION IN BA- NACH AND HILBERT SPACES. 2019. hal-02264895v2 HAL Id: hal-02264895 https://hal.archives-ouvertes.fr/hal-02264895v2 Preprint submitted on 19 Dec 2019 (v2), last revised 9 Jul 2020 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. GALERKIN APPROXIMATION IN BANACH AND HILBERT SPACES W. ARENDT, I. CHALENDAR, AND R. EYMARD Abstract. In this paper we study the conforming Galerkin ap- proximation of the problem: find u such that a(u, v)= L, v for all v , where and are Hilbert∈U or Banach spaces, ha is ai continuous∈ V bilinear orU sesquilinearV form and L ′ a given data. The approximate solution is sought in a finite dimensional∈ V subspace of , and test functions are taken in a finite dimensional subspace of U. We provide a necessary and sufficient condition on the form a forV convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint problem. We also characterize the fact that has a finite dimensional Schauder decomposition in terms of propertiesU related to the Galerkin approximation. -
Topology Proceedings
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 26, 2001{2002 Pages 695{707 WEAKLY EBERLEIN COMPACT SPACES DANIEL JARDON´ ∗ Abstract. Call a space X weakly splittable if, for each f X 2 R , there exists a σ-compact F Cp(X) such that f F (the bar denotes the closure in RX⊂). A weakly splittable com-2 pact space is called weakly Eberlein compact. We prove that weakly Eberlein compact spaces have almost the same prop- erties as Eberlein compact spaces. We show that any weakly Eberlein compact space of cardinality 6 c is Eberlein com- pact. We prove that a compact space X is weakly Eberlein compact if and only if X is splittable over the class of Eber- lein compact spaces and that every countably compact weakly splittable space has the Preiss{Simon property. 0. Introduction The first one to study weakly compact subspaces of Banach spaces was Eberlein [Eb]. His results showed that these compact spaces are very important and have numerous applications in many areas of mathematics. That is why they were called Eberlein com- pact spaces. In fact, a compact space X is Eberlein compact if and only if Cp(X) has a σ-compact dense subspace and it is a non-trivial theorem that these two definitions are equivalent. The basic results of the theory of Eberlein compact spaces have many 2000 Mathematics Subject Classification. -
Locally Solid Riesz Spaces with Applications to Economics / Charalambos D
http://dx.doi.org/10.1090/surv/105 alambos D. Alipr Lie University \ Burkinshaw na University-Purdue EDITORIAL COMMITTEE Jerry L. Bona Michael P. Loss Peter S. Landweber, Chair Tudor Stefan Ratiu J. T. Stafford 2000 Mathematics Subject Classification. Primary 46A40, 46B40, 47B60, 47B65, 91B50; Secondary 28A33. Selected excerpts in this Second Edition are reprinted with the permissions of Cambridge University Press, the Canadian Mathematical Bulletin, Elsevier Science/Academic Press, and the Illinois Journal of Mathematics. For additional information and updates on this book, visit www.ams.org/bookpages/surv-105 Library of Congress Cataloging-in-Publication Data Aliprantis, Charalambos D. Locally solid Riesz spaces with applications to economics / Charalambos D. Aliprantis, Owen Burkinshaw.—2nd ed. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 105) Rev. ed. of: Locally solid Riesz spaces. 1978. Includes bibliographical references and index. ISBN 0-8218-3408-8 (alk. paper) 1. Riesz spaces. 2. Economics, Mathematical. I. Burkinshaw, Owen. II. Aliprantis, Char alambos D. III. Locally solid Riesz spaces. IV. Title. V. Mathematical surveys and mono graphs ; no. 105. QA322 .A39 2003 bib'.73—dc22 2003057948 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed
Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed To cite this version: Nasiruddin Ahmed. Weak Compactness in the Space of Operator Valued Measures and Optimal Control. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. pp.49-58, 10.1007/978-3-642-36062-6_5. hal-01347522 HAL Id: hal-01347522 https://hal.inria.fr/hal-01347522 Submitted on 21 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License WEAK COMPACTNESS IN THE SPACE OF OPERATOR VALUED MEASURES AND OPTIMAL CONTROL N.U.Ahmed EECS, University of Ottawa, Ottawa, Canada Abstract. In this paper we present a brief review of some important results on weak compactness in the space of vector valued measures. We also review some recent results of the author on weak compactness of any set of operator valued measures. These results are then applied to optimal structural feedback control for deterministic systems on infinite dimensional spaces. Keywords: Space of Operator valued measures, Countably additive op- erator valued measures, Weak compactness, Semigroups of bounded lin- ear operators, Optimal Structural control. -
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. -
Dentability and Extreme Points in Banach Spaces*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 16, 78-90 (1974) Dentability and Extreme Points in Banach Spaces* R. R. PHELPS University of Washington, Seattle, Washington 98195 Communicated by R. Phillips Received November 20, 1973 It is shown that a Banach space E has the Radon-Nikodym property (equivalently, every bounded subset of E is dentable) if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points. Using recent work of Namioka, some analogous results are obtained concerning weak* strongly exposed points of weak* compact convex subsets of certain dual Banach spaces. The notion of a dentable subset of a Banach space was introduced by Rieffel[16] in conjunction with a Radon-Nikodym theorem for Banach space-valued measures. Subsequent work by Maynard [12] and by Davis and Phelps [6] (also by Huff [S]) has shown that those Banach spaces in which Rieffel’s Radon-Nikodym theorem is valid are precisely the ones in which every bounded closed convex set is dentable (definition below). Diestel [7] observed that the classes of spaces (e.g., reflexive spaces, separable conjugate spaces) which are known to have this property (the “Radon-Nikodym property”) appear to coincide with those which are known to have the property that every bounded closed convex subset is the closed convex hull of its extreme points (the “Krein-Milman property”) and he raised the question as to whether the two properties are equivalent. -
Extreme Points of Vector Functions
EXTREME POINTS OF VECTOR FUNCTIONS SAMUEL KARLIN Several previous investigations have appeared in the literature which discuss the nature of the set T in »-dimensional space spanned by the vectors ifsdui, ■ ■ ■ , fsdp.n) where 5 ranges over all measur- able sets. It was first shown by Liapounov that xi ui, • • ■ , un are atomless measures, then the set F is convex and closed. Extensions of this result were achieved by D. Blackwell [l] and Dvoretsky, Wald, and Wolfowitz [2]. A study of the extreme points in certain function spaces is made from which the theorems concerning the range of finite vector measures are deduced as special cases. However, the extreme point theorems developed here have independent interest. In addition, some results dealing with infinite direct products of func- tion spaces are presented. Many of these ideas have statistical in- terpretation and applications in terms of replacing randomized tests by nonrandomized tests. 1. Preliminaries. Let p. denote a finite measure defined on a Borel field of sets $ given on an abstract set X. Let Liu, 5) denote the space of all integrable functions with respect to u. Finally, let Miß, %) be the space of all essentially bounded measurable functions defined on X. It is well known that Af(/i, 5) constitutes the conjugate space of Liu, %) and consequently the unit sphere of Af(/x, %) is bi- compact in the weak * topology [3]. Let (®Ln) denote the direct product of L taken with itself » times and take i®Mn) as the direct product of M n times. With appropriate choice of norm, i®M") becomes the conjugate space to (®7,n). -
Arxiv:1705.02625V1 [Math.FA] 7 May 2017 in H Ugra Ainlsinicfn Ne Rn DFNI-I02/10
STRONGLY EXTREME POINTS AND APPROXIMATION PROPERTIES TROND A. ABRAHAMSEN, PETR HAJEK,´ OLAV NYGAARD, AND STANIMIR TROYANSKI Abstract. We show that if x is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approxi- mation at x, then x is already a denting point. It turns out that such an approximation of the identity exists at any strongly ex- treme point of the unit ball of a Banach space with the uncondi- tional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also construct a non-symmetric norm on c0 for which all points on the unit sphere are strongly extreme, but none of these points are denting. 1. Introduction Let X be a (real) Banach space and denote by BX its unit ball, SX its unit sphere, and X∗ its topological dual. Let A be a non-empty set in X. By a slice of A we mean a subset of A of the form S(A, x∗,ε) := x A : x∗(x) > M ε { ∈ − } ∗ ∗ ∗ ∗ where ε > 0, x X with x = 0, and M = supx∈A x (x). We will simply write S(x∗∈,ε) for a slice of6 a set when it is clear from the setting what set we are considering slices of. Definition 1.1. Let B be a non-empty bounded closed convex set in a Banach space X and let x B. -
Grothendieck Spaces with the Dunford–Pettis Property
GROTHENDIECK SPACES WITH THE DUNFORD–PETTIS PROPERTY ANGELA A. ALBANESE*, JOSÉ BONET+ AND WERNER J. RICKER Abstract. Banach spaces which are Grothendieck spaces with the Dunford– Pettis property (briefly, GDP) are classical. A systematic treatment of GDP– Fréchet spaces occurs in [12]. This investigation is continued here for locally convex Hausdorff spaces. The product and (most) inductive limits of GDP– space are again GDP–spaces. Also, every complete injective space is a GDP– space. For p ∈ {0} ∪ [1, ∞) it is shown that the classical co–echelon spaces kp(V ) and Kp(V ) are GDP–spaces if and only if they are Montel. On the other hand, K∞(V ) is always a GDP–space and k∞(V ) is a GDP–space whenever its (Fréchet) predual, i.e., the Köthe echelon space λ1(A), is distinguished. 1. Introduction. Grothendieck spaces with the Dunford–Pettis property (briefly, GDP) play a prominent role in the theory of Banach spaces and vector measures; see Ch.VI of [17], especially the Notes and Remarks, and [18]. Known examples include L∞, ∞ ∞ H (D), injective Banach spaces (e.g. ` ) and certain C(K) spaces. D. Dean showed in [14] that a GDP–space does not admit any Schauder decomposition; see also [26, Corollary 8]. This has serious consequences for spectral measures in such spaces, [31]. For non–normable spaces the situation changes dramatically. Every Fréchet Montel space X is a GDP–space, [12, Remark 2.2]. Other than Montel spaces, the only known non–normable Fréchet space which is a GDP–space is the Köthe echelon space λ∞(A), for an arbitrary Köthe matrix A, [12, Proposition 3.1]. -
Free Banach Spaces and the Approximation Properties
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 5, May 2014, Pages 1681–1687 S 0002-9939(2014)11933-2 Article electronically published on February 18, 2014 FREE BANACH SPACES AND THE APPROXIMATION PROPERTIES GILLES GODEFROY AND NARUTAKA OZAWA (Communicated by Thomas Schlumprecht) Abstract. We characterize the metric spaces whose free spaces have the bounded approximation property through a Lipschitz analogue of the local reflexivity principle. We show that there exist compact metric spaces whose free spaces fail the approximation property. 1. Introduction Let M be a pointed metric space, that is, a metric space equipped with a distin- guished point denoted 0. We denote by Lip0(M) the Banach space of all real-valued Lipschitz functions defined on M which vanish at 0, equipped with the natural Lip- schitz norm |f(x) − f(y)| f =sup ;(x, y) ∈ M 2,x= y . L x − y ∈ For all x M, the Dirac measure δ(x) defines a continuous linear form on Lip0(M). Equicontinuity shows that the closed unit ball of Lip0(M) is compact for pointwise { ∈ } ∗ convergence on M, and thus the closed linear span of δ(x); x M in Lip0(M) is an isometric predual of Lip0(M). This predual is called the Arens-Eells space of M in [We] and (when M is a Banach space) the Lipschitz-free space over M in [GK], denoted by F(M). We will use this notation and simply call F(M)thefree space over M.WhenM is separable, the Banach space F(M) is separable as well, { ∈ } ∗ since the set δ(x); x M equipped with the distance induced by Lip0(M) is isometric to M. -
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.