Tensor Products of Vector Seminormed Spaces
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The Semi-M Property for Normed Riesz Spaces Compositio Mathematica, Tome 34, No 2 (1977), P
COMPOSITIO MATHEMATICA EP DE JONGE The semi-M property for normed Riesz spaces Compositio Mathematica, tome 34, no 2 (1977), p. 147-172 <http://www.numdam.org/item?id=CM_1977__34_2_147_0> © Foundation Compositio Mathematica, 1977, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. 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/ COMPOSITIO MATHEMATICA, Vol. 34, Fasc. 2, 1977, pag. 147-172 Noordhoff International Publishing Printed in the Netherlands THE SEMI-M PROPERTY FOR NORMED RIESZ SPACES Ep de Jonge 1. Introduction It is well-known that if (0394, F, IL) is a u-finite measure space and if 1 ~ p 00, then the Banach dual L *p of the Banach space Lp = Lp(0394, IL) can be identified with Lq = Lq(L1, 03BC), where p-1 + q-1 = 1. For p =00 the situation is different; the space Li is a linear subspace of L*, and only in a very trivial situation (the finite-dimensional case) we have Li = Lfi. Restricting ourselves to the real case, the Banach dual L *~ is a (real) Riesz space, i.e., a vector lattice, and Li is now a band in L*. The disjoint complement (i.e., the set of all elements in L* disjoint to all elements in LI) is also a band in L*, called the band of singular linear functionals on Loo. -
A Note on Bilinear Forms
A note on bilinear forms Darij Grinberg July 9, 2020 Contents 1. Introduction1 2. Bilinear maps and forms1 3. Left and right orthogonal spaces4 4. Interlude: Symmetric and antisymmetric bilinear forms 14 5. The morphism of quotients I 15 6. The morphism of quotients II 20 7. More on orthogonal spaces 24 1. Introduction In this note, we shall prove some elementary linear-algebraic properties of bi- linear forms. These properties generalize some of the standard facts about non- degenerate bilinear forms on finite-dimensional vector spaces (e.g., the fact that ? V? = V for a vector subspace V of a vector space A equipped with a nonde- generate symmetric bilinear form) to bilinear forms which may be degenerate. They are unlikely to be new, but I have not found them explored anywhere, whence this note. 2. Bilinear maps and forms We fix a field k. This field will be fixed for the rest of this note. The notion of a “vector space” will always be understood to mean “k-vector space”. The 1 A note on bilinear forms July 9, 2020 word “subspace” will always mean “k-vector subspace”. The word “linear” will always mean “k-linear”. If V and W are two k-vector spaces, then Hom (V, W) denotes the vector space of all linear maps from V to W. If S is a subset of a vector space V, then span S will mean the span of S (that is, the subspace of V spanned by S). We recall the definition of a bilinear map: Definition 2.1. -
Vector Lattice Covers of Ideals and Bands in Pre-Riesz Spaces
Vector lattice covers of ideals and bands in pre-Riesz spaces Anke Kalauch,∗ Helena Malinowski† November 12, 2018 Abstract Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover Y for a pre-Riesz space X, we address the question how to find vector lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a vector lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y . 1 Introduction In the analysis of partially ordered vector spaces, subspaces as ideals and bands play a central role. In vector lattices, problems involving those subspaces are broadly discussed in the literature, see e.g. [1, 15, 22]. Directed ideals in partially ordered vector spaces were introduced in [6, Definition 2.2] and studied in [2, 3, 14, 5]; for a more recent overview see [12]. The investigation of disjointness and bands in partially ordered vector spaces starts in [9], where for the definition of disjointness sets of upper bounds are used instead of lattice operations. Here a band is defined to be a set that equals its double-disjoint complement, analogously to the notion in arXiv:1801.07191v2 [math.FA] 8 Nov 2018 Archimedean vector lattices. Ideals and bands are mostly considered in pre-Riesz spaces. -
Bornologically Isomorphic Representations of Tensor Distributions
Bornologically isomorphic representations of distributions on manifolds E. Nigsch Thursday 15th November, 2018 Abstract Distributional tensor fields can be regarded as multilinear mappings with distributional values or as (classical) tensor fields with distribu- tional coefficients. We show that the corresponding isomorphisms hold also in the bornological setting. 1 Introduction ′ ′ ′r s ′ Let D (M) := Γc(M, Vol(M)) and Ds (M) := Γc(M, Tr(M) ⊗ Vol(M)) be the strong duals of the space of compactly supported sections of the volume s bundle Vol(M) and of its tensor product with the tensor bundle Tr(M) over a manifold; these are the spaces of scalar and tensor distributions on M as defined in [?, ?]. A property of the space of tensor distributions which is fundamental in distributional geometry is given by the C∞(M)-module isomorphisms ′r ∼ s ′ ∼ r ′ Ds (M) = LC∞(M)(Tr (M), D (M)) = Ts (M) ⊗C∞(M) D (M) (1) (cf. [?, Theorem 3.1.12 and Corollary 3.1.15]) where C∞(M) is the space of smooth functions on M. In[?] a space of Colombeau-type nonlinear generalized tensor fields was constructed. This involved handling smooth functions (in the sense of convenient calculus as developed in [?]) in par- arXiv:1105.1642v1 [math.FA] 9 May 2011 ∞ r ′ ticular on the C (M)-module tensor products Ts (M) ⊗C∞(M) D (M) and Γ(E) ⊗C∞(M) Γ(F ), where Γ(E) denotes the space of smooth sections of a vector bundle E over M. In[?], however, only minor attention was paid to questions of topology on these tensor products. -
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. -
A Symplectic Banach Space with No Lagrangian Subspaces
transactions of the american mathematical society Volume 273, Number 1, September 1982 A SYMPLECTIC BANACHSPACE WITH NO LAGRANGIANSUBSPACES BY N. J. KALTON1 AND R. C. SWANSON Abstract. In this paper we construct a symplectic Banach space (X, Ü) which does not split as a direct sum of closed isotropic subspaces. Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form Y X Y* is settled in the negative. The proof also shows that £(A") admits a nontrivial continuous homomorphism into £(//) where H is a Hilbert space. 1. Introduction. Given a Banach space E, a linear symplectic form on F is a continuous bilinear map ß: E X E -> R which is alternating and nondegenerate in the (strong) sense that the induced map ß: E — E* given by Û(e)(f) = ü(e, f) is an isomorphism of E onto E*. A Banach space with such a form is called a symplectic Banach space. It can be shown, by essentially the argument of Lemma 2 below, that any symplectic Banach space can be renormed so that ß is an isometry. Any symplectic Banach space is reflexive. Standard examples of symplectic Banach spaces all arise in the following way. Let F be a reflexive Banach space and set E — Y © Y*. Define the linear symplectic form fiyby Qy[(^. y% (z>z*)] = z*(y) ~y*(z)- We define two symplectic spaces (£,, ß,) and (E2, ß2) to be equivalent if there is an isomorphism A : Ex -» E2 such that Q2(Ax, Ay) = ß,(x, y). A. Weinstein [10] has asked the question whether every symplectic Banach space is equivalent to one of the form (Y © Y*, üy). -
Self-Bilinear Map from One Way Encoding System and Indistinguishability Obfuscation
Self-bilinear Map from One Way Encoding System and Indistinguishability Obfuscation Huang Zhang1,4, Fangguo Zhang1,4, Baodian Wei2,4, and Yusong Du3 1 School of Information Science and Technology, Sun Yat-sen University 2 School of Data and Computer Science, Sun Yat-sen University 3 School of Information Management, Sun Yat-sen University 4 Guangdong Key Laboratory of Information Security Technology Guangzhou 510006, China [email protected] Abstract. The bilinear map whose domain and target sets are identical is called the self-bilinear map. Original self-bilinear maps are defined over cyclic groups. This brings a lot of limitations to construct secure self-bilinear schemes. Since the map itself reveals information about the underlying cyclic group, hardness assumptions on DDHP and CDHP may not hold any more. In this paper, we used iO to construct a self-bilinear map from generic sets. These sets should own several properties. A new notion, One Way Encoding System (OWES), is proposed to describe formally the properties those sets should hold. An Encoding Division Problem is defined to complete the security proof. As an instance of the generic construction, we propose a concrete scheme built on the GGH graded encoding system and state that any 1-graded encoding system may satisfy the requirements of OWES. Finally, we discuss the hardness of EDP in the GGH graded encoding system. Keywords: self-bilinear map, multi-linear map, indistinguishability obfus- cation, one way encoding system. 1 Introduction The bilinear map is a very useful cryptographic primitive. It provides solutions for many cryptographic applications such as identity-based encryptions [2], non- interactive zero-knowledge proof systems [16], attribute-based encryptions [21] and short signatures [3, 23], etc. -
Version of 4.9.09 Chapter 35 Riesz Spaces the Next Three Chapters Are
Version of 4.9.09 Chapter 35 Riesz spaces The next three chapters are devoted to an abstract description of the ‘function spaces’ described in Chapter 24, this time concentrating on their internal structure and relationships with their associated measure algebras. I find that any convincing account of these must involve a substantial amount of general theory concerning partially ordered linear spaces, and in particular various types of Riesz space or vector lattice. I therefore provide an introduction to this theory, a kind of appendix built into the middle of the volume. The relation of this chapter to the next two is very like the relation of Chapter 31 to Chapter 32. As with Chapter 31, it is not really meant to be read for its own sake; those with a particular interest in Riesz spaces might be better served by Luxemburg & Zaanen 71, Schaefer 74, Zaanen 83 or my own book Fremlin 74a. I begin with three sections in an easy gradation towards the particular class of spaces which we need to understand: partially ordered linear spaces (§351), general Riesz spaces (§352) and Archimedean Riesz spaces (§353); the last includes notes on Dedekind (σ-)complete spaces. These sections cover the fragments of the algebraic theory of Riesz spaces which I will use. In the second half of the chapter, I deal with normed Riesz spaces (in particular, L- and M-spaces)(§354), spaces of linear operators (§355) and dual Riesz spaces (§356). Version of 16.10.07 351 Partially ordered linear spaces I begin with an account of the most basic structures which involve an order relation on a linear space, partially ordered linear spaces. -
RIESZ SPACES " Given by Prof
EUR 3140.Θ ÄO!·'· 1 \n\< tl '*·»ΤΗΗΊ)ίβ§"ίί IJM llHi 'li»**. 'tí fe É!>?jpfc EUROPEAN ATOMIC ENERGY COMMUNITY EURATOM LECTURES ON " RIESZ SPACES " given by Prof. A. C. ZAANEN iiil^{TT"1! i ■ BIT' '.'I . * . Ι'βΗΤ Editor : R. F. GLODEN !!i>M!?ÄÉ«il Siffifi 1966 loint Nuclear Research Center Ispra Establishment - Italy Scientific Information Processing Center - CETIS »»cm*if'-WW;flW4)apro 'BfiW»¡kHh;i.u·^: 2Λ*.;;, tf! :1Γ:«Μ$ ■ ■ J'ÎHO *sUr! if nb-. I;·"'-ii ;Γ^*Ε»"^Β1 hiik*MW!?5?'J.-i,K^ fill"; UP» LEGAL NOTICE This document was prepared under the sponsorship of the Commission of the European Atomic Energy Community (EURATOM). Neither the EURATOM Commission, its contractors nor any person acting- on their behalf : Make any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this document, or that the use of any information, apparatus, method, or process disclosed in this document may not infringe privately owned rights ; or Assume any liability with respect to the use of, or for damages resulting from the use of any information, apparatus, method or process disclosed in this document. This report is on sale at the addresses listed on cover page 4 at the price of FF 8.50 FB 85 DM 6.80 Lit. 1060 Fl. 6.20 When ordering, please quote the EUR number and the titl which are indicated on the cover of each report. m ■pejs!« EUR 3140.e LECTURES ON « RIESZ SPACES » given by Prof. A.C. -
A Bilinear Version of Orlicz-Pettis Theorem. a Bilinear Version of Orlicz-Pettis Theorem
A bilinear version of Orlicz-Pettis theorem. O. Blasco 1 Department of Mathematics. Universitat de Val`encia.Burjassot 46100 (Val`encia), Spain. J.M. Calabuig ∗,2 Instituto Universitario de Matem´atica Pura y Aplicada. Universitat Polit`ecnica de Val`encia. Val`encia46022, Spain. T. Signes 3 Department of Applied Mathematics. Universidad de Murcia. Espinardo 30100 (Murcia), Spain. A bilinear version of Orlicz-Pettis theorem. O. Blasco Department of Mathematics. Universitat de Val`encia.Burjassot 46100 (Val`encia), Spain. J.M. Calabuig Instituto Universitario de Matem´atica Pura y Aplicada. Universitat Polit`ecnica de Val`encia. Val`encia46022, Spain. T. Signes Department of Applied Mathematics. Universidad de Murcia. Espinardo 30100 (Murcia), Spain. Abstract Given three Banach spaces X, Y and Z and a bounded bilinear map B : X×Y → Z, P∞ a sequence x = (xn)n ⊆ X is called B-absolutely summable if n=1 kB(xn, y)kZ < 1 ∞ for any y ∈ Y . Connections of this space with `weak(X) are presented. A sequence P∞ ∗ x = (xn)n ⊆ X is called B-unconditionally summable if n=1 |hB(xn, y), z i| < ∞ Preprint submitted to Elsevier 21 December 2007 ∗ ∗ for any y ∈ Y and z ∈ Z and for any M ⊆ N there exists xM ∈ X for which P ∗ ∗ ∗ ∗ n∈M hB(xn, y), z i = hB(xM , y), z i for all y ∈ Y and z ∈ Z . A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented. Key words: 40F05 Absolute and strong summability, 46A45 Sequence spaces, 46B45 Banach sequence spaces. 1 Notation and preliminaries. -
Integration of Functions with Values in a Riesz Space
Integration of functions with values in a Riesz space. Willem van Zuijlen Supervisor: Second reader: A.C.M. van Rooij O.W. van Gaans July 2012 Contents Introduction 2 Conventions and Notations 4 1 Preliminaries 7 1.1 The Riesz dual of a Riesz space . .7 1.2 Integrals for functions with values in a Banach space . 12 1.2.1 The Bochner integral . 12 1.2.2 The Pettis integral . 16 2 Integrals for functions with values in a Riesz space 17 2.1 The Bochner integral on Riesz space valued functions . 18 2.2 σ-simple functions . 20 2.3 The R-integral on Riesz space valued functions. 29 2.4 The U-integral on Riesz space valued functions . 37 2.5 The Pettis integral on Riesz space valued functions . 42 3 Comparing the integrals 47 3.1 Comparing the R-integral with the Bochner integral . 47 3.2 Comparing the U-integral with the R-integral . 49 3.3 Comparing the U-integral with the Bochner integral . 50 3.4 Comparing the U-integral with the strong and weak Pettis integral 51 3.5 Comparing the R-integral with the weak Pettis integral . 52 3.6 Comparing the R-integral with the strong Pettis integral . 54 4 Further properties of the R-integral 57 5 Examples 69 References 75 Index 76 1 Introduction In this thesis one will find definitions of integrals for functions with values in a Riesz space. The idea for this subject started when I began to learn about the Bochner integral (which is an integral for functions with values in a Banach space) and about Riesz spaces. -
The Order Convergence Structure
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Indagationes Mathematicae 21 (2011) 138–155 www.elsevier.com/locate/indag The order convergence structure Jan Harm van der Walt∗ Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa Received 20 September 2010; received in revised form 24 January 2011; accepted 9 February 2011 Communicated by Dr. B. de Pagter Abstract In this paper, we study order convergence and the order convergence structure in the context of σ- distributive lattices. Particular emphasis is placed on spaces with additional algebraic structure: we show that on a Riesz algebra with σ-order continuous multiplication, the order convergence structure is an algebra convergence structure, and construct the convergence vector space completion of an Archimedean Riesz space with respect to the order convergence structure. ⃝c 2011 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. All rights reserved. MSC: 06A19; 06F25; 54C30 Keywords: Order convergence; Convergence structure; Riesz space 1. Introduction A useful notion of convergence of sequences on a poset L is that of order convergence; see for instance [2,7]. Recall that a sequence .un/ on L order converges to u 2 L whenever there is an increasing sequence (λn/ and a decreasing sequence (µn/ on L such that sup λn D u D inf µn and λn ≤ un ≤ µn; n 2 N: (1) n2N n2N In case the poset L is a Riesz space, the relation (1) is equivalent to the following: there exists a sequence (λn/ that decreases to 0 such that ju − unj ≤ λn; n 2 N: (2) ∗ Tel.: +27 12 420 2819; fax: +27 12 420 3893.