DOCSLIB.ORG

Stability Constants and the Homology of Quasi-Banach Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Stability Constants and the Homology of Quasi-Banach Spaces Stability constants and the homology of quasi-Banach spaces F¶elixCabello S¶anchez and Jes¶usM. F. Castillo Abstract. We a±rmatively solve the main problems posed by Laczkovich and Paulin in Stability constants in linear spaces, Constructive Approximation 34 (2011) 89{106 (do there exist cases in which the second Whitney constant is ¯nite while the approximation constant is in¯nite?) and by Cabello and Castillo in The long homol- ogy sequence for quasi-Banach spaces, with applications, Positivity 8 (2004) 379{394 (do there exist Banach spaces X; Y for which Ext(X; Y ) is Hausfdor® and non-zero?). In fact, we show that these two problems are the same. AMS (2010) Class. Number. 41A30, 46B20, 46M15, 46M18, 46A16, 26B25, 52A05, 46B08. 1. Introduction In this paper we solve the main problems posed [2] and in [11]. This introductory section is devoted to explain the problems and the solutions we present. The reader is addressed to Section 2 for all unexplained terms and notation. The underlying idea of [2] was to study the spaces Ext(X; Y ) of exact sequences of quasi-Banach spaces 0 ! Y ! Z ! X ! 0 by means of certain nonlinear maps F : X ! Y called quasilinear maps. In that paper we introduced (semi-quasi-normed) linear topologies in the spaces Ext(X; Y ). Such topologies are completely natural in this setting since they make continuous all the linear maps appearing in the associ- ated homology sequences. We encountered considerable di±culties in studying the Hausdor® character of Ext(X; Y ) for arbitrary spaces X; Y , although we managed to produce some pairs of quasi-Banach spaces X and Y for which Ext(X; Y ) is (nonzero and) Hausdor®. Those examples invariably required that L(X; Y ) = 0, which is im- possible in a Banach space context. Thus the main problem left open in [2, p. 387] was to know if there are Banach spaces X and Y for which Ext(X; Y ) is nonzero and This research has been supported in part by project MTM2010-20190-C02-01 and the program Junta de Extremadura GR10113 IV Plan Regional I+D+i, Ayudas a Grupos de Investigaci¶on. 1 2 FELIX¶ CABELLO SANCHEZ¶ AND JESUS¶ M. F. CASTILLO Hausdor®. We will show in this paper that this is indeed the case. This paper also solves the main problem in [11]. In fact, it was this problem who suggested us the approach of the present paper and the \form" that the examples in Section 6 had to have. In [11] Laczkovich and Paulin studied the stability of approx- imately a±ne and Jensen functions on convex sets of linear spaces. They show that, in many respects, it is enough to work the case where the convex set is the unit ball of a Banach space. Reduced to its basic elements the main problem in [11] can be stated as follows. Let B a convex set and Y a Banach space. A function f : B ! Y is approximately a±ne if kf(tx + (1 ¡ t)y) ¡ tf(x) ¡ (1 ¡ t)f(y)kY · 1 for every x; y 2 D and every t 2 [0; 1]. Suppose there is a constant A0 = A0(B; Y ) such that for every bounded approximately a±ne function f : B ! Y there is a true a±ne function a : B ! Y such that kf(x) ¡ a(x)k · A0 for every x 2 B. Does it follow that for every (in general unbounded) approximately a±ne function f : B ! Y there is an a±ne function a such that supx2B kf(x) ¡ a(x)k < 1? We show in Section 8 that the Examples in Section 6 provide a negative answer to this problem. The idea is that if B is the unit ball of X, then every approximately a±ne function f : B ! Y is (up to a bounded perturbation) the restriction of some quasilinear map F : X ! Y . Thus A0(B; Y ) and K0(X; Y ) are nearly proportional and so, it su±ces to get a pair of Banach spaces with K0(X; Y ) ¯nite but admitting a nontrivial quasilinear map F : X ! Y . Let us describe the organization of the paper. Section 2 is preliminary: it contains the necessary background on quasilinear maps and extensions. In Section 3 we give a seemingly new representation of Ext(X; Y ), we denote by Ext$(X; Y ), which is based on a relatively projective presentation of X and carries a natural topology. Section 4 is devoted to show that the \old" functor Ext and the \new" functor Ext$ are naturally equivalent, both algebraically and topologically. In Section 5 we give the basic criterion to determine when Ext(X; Y ) is Hausdor®. It turns out that Ext(X; Y ) is Hausdor® if and only if a certain parameter K0(X; Y ) (depending only on the behavior of the bounded quasilinear maps F : X ! Y ) is ¯nite. In Section 6 we arrive to the main counter-examples: we show that Ext(X; Y ) is nonzero and Hausdor® if, for instance, X = `p(I) with 1 < p < 1 and I uncountable and Y = c0 is the space of all null sequences with the sup norm. The results of Section 7 con¯rm that Ext(X; Y ) has a quite strong tendency to not being Hausdor®. The paper has been organized so that the interested reader can go straight to the solution of Laczkovich-Paulin problem. The shortest path is to read ¯rst Sections 2.2 and 2.3, then Section 6 and ¯nally Section 8. STABILITY CONSTANTS AND THE HOMOLOGY OF QUASI-BANACH SPACES 3 2. Background on quasilinear maps and extensions of quasi-Banach spaces 2.1. Notation. Let X be a (real or complex) linear space. A semi quasi-norm is a function % : X ! R+ satisfying the following conditions: (a) %(¸x) = j¸j%(x) for every x 2 X and every scalar ¸. (b) There is a constant ¢ such that %(x + y) · ¢(%(x) + %(y)) for every x; y 2 X. A semi quasi-normed space is a linear space furnished X with a semi quasi-norm % which gives rise to a linear topology, namely the least linear topology for which the set BX = fx 2 X : %(x) · 1g is a neighborhood of the origin. A semi quasi-Banach space is a complete semi quasi-normed space. If %(x) = 0 implies x = 0, then % is said to be a quasi-norm and X is Hausdor®. Throughout the paper we denote by ¢% (or ¢X if there is no risk of confusion) the modulus of concavity of (X; %), that is, the least constant ¢ for which (b) holds. Let X and Y be quasi-Banach spaces. A mapping F : X ! Y is said to be homogeneous if F (¸x) = ¸F (x) for every scalar ¸ and every x 2 X. A homogeneous map F is bounded if there is a constant C such that kF (x)kY · CkxkX for all x 2 X; equivalently if it is uniformly bounded on the unit ball. We write kF k for the least constant C for which the preceding inequality holds. Clearly, kF k = supkxk·1 kF (x)k. This is coherent with the standard notation for the (quasi-) norm of a linear operator. 2.2. Quasilinear maps. Let X and Y be quasi-Banach spaces. A map F : X ! Y is said to be quasilinear if it is homogeneous and satis¯es an estimate (1) kF (x + y) ¡ F (x) ¡ F (y)kY · Q (kxkX + kykX ) for some constant Q and all x; y 2 X. The smallest constant Q above shall be denoted by Q(F ). All linear maps, continuous or not, are quasilinear and so are the bounded (homo- geneous) maps. Let us say that a quasilinear map F is trivial if it is the sum of a linear map L and a bounded homogeneous map B. This happens if and only F is at ¯nite \distance" from L in the sense that kF ¡ Lk < 1. Let us introduce the \approximation constants" for quasilinear maps as follows. Definition 1. Given quasi-Banach spaces X and Y we denote by K0(X; Y ) the in¯mum of those constants C for which the following statement is true: if F : X ! Y is a bounded quasilinear map, then there is a linear map L : X ! Y such that kF ¡ Lk · CQ(F ). We de¯ne K(X; Y ) analogously just omitting the word `bounded'. Observe that K0(X; Y ) does not vary if we replace \bounded" by \trivial" in the de¯nition. 4 FELIX¶ CABELLO SANCHEZ¶ AND JESUS¶ M. F. CASTILLO 2.3. Twisted sums. The link between quasilinear maps and the homology of (quasi-) Banach spaces is the following construction, due to Kalton [6] and Ribe [13]. Suppose F : X ! Y is quasilinear. We consider the product space Y £ X and we furnish it with the following quasinorm: k(y; x)kF = ky ¡ F (x)kY + kxkX : Following a long standing tradition, we denote the resulting quasi-Banach space by Y ©F X. It is obvious that the operator { : Y ! Y ©F X sending y to (y; 0) is an isometric embedding, so we may regard Y as a subspace of Y ©F X. The connection between X and Y ©F X is less obvious. Consider the operator ¼ : Y ©F X ! X de¯ned by ¼(y; x) = x. We have k¼k · 1, by the very de¯nition of the quasi-norm in Y ©F X. Besides, ¼ maps the unit ball of Y ©F X onto the unit ball of X since ¼(F (x); x) = x and k(F (x); x)kF = kxkX .
Load more

Recommended publications
  • PROBLEM SHEET-I IST, Functional Analysis, Bhaskaracharya Pratishthana, Pune (Oct.23-Nov.4, 2017)
    PROBLEM SHEET-I IST, Functional Analysis, Bhaskaracharya Pratishthana, Pune (Oct.23-Nov.4, 2017) 1. Given a finite family of Banach spaces , show that the product {푋푖; 1 ≤ 푖 ≤ } 푛 푋 = ∏ 푋푖 = {푥 = 푥푖푖=1; 푥푖 ∈ 푋푖,1 ≤ 푖 ≤ } is a Banach space when푖 =equipped 1 with the norm: 푛 푛 ‖푥‖ = ∑‖푥푖‖ , 푥 = 푥푖푖=1 ∈ 푋. 2. Show that a normed space X is complete푖=1 if and only if absolutely convergent series in X are convergent. 3. Compute in C[0, 1] where ‖ + ‖ 4. Show that it is possible that two푥 = metrics 푠푖휋푥, on 푥a set= 푐푠휋푥.are equivalent whereas one of them is complete and that the other is incomplete. 5. Show that the closure of a subspace/convex subset of a normed space is a subspace/convex subset. How about the convex hull of a closed set? 6. Prove that a linear functional on a normed space is continuous if and only Kerf is a closed subspace of X. 7. For show that 1 ≤ < < ∞, ℓ ⊊ ℓ . PROBLEM SHEET-II IST, Functional Analysis, Bhaskaracharya Pratishthana, Pune (Oct.23-Nov.4, 2017) 1. Give an example of a discontinuous linear map between Banach spaces. 2. Prove that two equivalent norms are complete or incomplete together. 3. Show that a normed space is finite dimensional if all norms on it are equivalent. 4. Given that a normed space X admits a finite total subset of X*, can you say that it is finite dimensional? 5. What can you say about a normed space if it is topologically homeomorphic to a a finite dimensional normed space.
    [Show full text]
  • Introduction to Functional Analysis Part III, Autumn 2004
    Introduction to Functional Analysis Part III, Autumn 2004 T. W. KÄorner October 21, 2004 Small print This is just a ¯rst draft for the course. The content of the course will be what I say, not what these notes say. Experience shows that skeleton notes (at least when I write them) are very error prone so use these notes with care. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the ¯rst ¯nder of particular errors. Contents 1 Some notes on prerequisites 2 2 Baire category 4 3 Non-existence of functions of several variables 5 4 The principle of uniform boundedness 7 5 Zorn's lemma and Tychonov's theorem 11 6 The Hahn-Banach theorem 15 7 Banach algebras 17 8 Maximal ideals 21 9 Analytic functions 21 10 Maximal ideals 23 11 The Gelfand representation 24 1 12 Finding the Gelfand representation 26 13 Three more uses of Hahn-Banach 29 14 The Rivlin-Shapiro formula 31 1 Some notes on prerequisites Many years ago it was more or less clear what could and what could not be assumed in an introductory functional analysis course. Since then, however, many of the concepts have drifted into courses at lower levels. I shall therefore assume that you know what is a normed space, and what is a a linear map and that you can do the following exercise. Exercise 1. Let (X; k kX ) and (Y; k kY ) be normed spaces. (i) If T : X ! Y is linear, then T is continuous if and only if there exists a constant K such that kT xkY · KkxkX for all x 2 X.
    [Show full text]
  • Greedy Approximation for Biorthogonal Systems in Quasi-Banach Spaces
    GREEDY APPROXIMATION FOR BIORTHOGONAL SYSTEMS IN QUASI-BANACH SPACES FERNANDO ALBIAC, JOSE´ L. ANSORENA, PABLO M. BERNA,´ AND PRZEMYSLAW WOJTASZCZYK Abstract. The general problem addressed in this work is the development of a systematic study of the thresholding greedy al- gorithm for general biorthogonal systems (also known as Marku- shevich bases) in quasi-Banach spaces from a functional-analytic point of view. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (nontrivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new proper- ties arise, and the relations amongst them are carefully discussed. Introduction The subject of finding estimates for the rate of approximation of a function by means of essentially nonlinear algorithms with respect to biorthogonal systems and, in particular, the greedy approximation al- gorithm using bases, has attracted much attention for the last twenty years, on the one hand from researchers interested in the applied nature 2010 Mathematics Subject Classification. 46B15, 41A65. Key words and phrases. quasi-greedy basis, almost greedy , greedy basis, uncon- ditional basis, democratic basis, quasi-Banach spaces. F. Albiac acknowledges the support of the Spanish Ministry for Economy and Competitivity Grant MTM2016-76808-P for Operators, lattices, and structure of Banach spaces. J.L. Ansorena acknowledges the support of the Spanish Ministry for Economy and Competitivity Grant MTM2014-53009-P for An´alisisVectorial, Multilineal y Aplicaciones. The research of P. M. Bern´awas partially supported by the grants MTM-2016-76566-P (MINECO, Spain) and 20906/PI/18 from Fun- daci´onS´eneca(Regi´onde Murcia, Spain).
    [Show full text]
  • Exercises for the Lecture Functional Analysis I Exercise 13 (2+2=4
    SAARLAND UNIVERSITY DEPARTMENT OF MATHEMATICS Prof. Dr. J¨orgEschmeier M. Sc. Sebastian Toth Exercises for the lecture Functional Analysis I Winter term 2020/2021 Sheet 4 To be submitted until: Thursday, 03.12.2020, before the lecture Exercise 13 (2+2=4 Points) Let A; B ⊂ `1 be defined by 1 A = (an)n2N 2 ` ; a2n = 0 for all n 2 N ; 1 −n B = (bn)n2N 2 ` ; b2n = 2 b2n+1 for all n 2 N : Show that: (a) A and B are closed linear subspaces of `1. (b) A + B ⊂ `1 is not closed. 1 (Hint : Consider the sequences (xk)k2N, (yk)k2N in ` defined as xk = e1 + e3 + ::: + e2k+1 and yk = −1 −k e0 + 2 e2 + ::: + 2 e2k + xk for k 2 N, where ei = (δi;n)n2N (i 2 N):) Exercise 14 (1+2+2=5 Points) Let (E; k · k) be a normed space and let E0 ⊂ E be a closed linear subspace. Let E=E0 be the quotient vector space of E modulo E0. Write the elements of E=E0 as [x] = x + E0 2 E=E0. Show that: (a) k[x]k = inffkx − yk; y 2 E0g defines a norm on E=E0. (b) The quotient map q : E ! E=E0; q(x) = [x] is continuous linear and open. (c) If E is complete, then E=E0 is complete. (Hint : Show that each absolutely convergent series in E=E0 converges.) Exercise 15 (4 Points) Let E be an infinite-dimensional vector space over K = R or K = C. Show that there is a discontinuous linear map u: E ! K.
    [Show full text]
  • Lipschitz Mappings
    On Differentiability and Surjectivity of a-Lipschitz Mappings. JACoPo PEJS~C~OWlCZ - ALF0~S0 VIG:~0LI (Genova) (*) (**) Summary. - In the ]irst part o] this paper the well known result that the Erdchet di]]erential and the asymptotic derivative o] a compact mapping is compact, is extended to ~-Lipschitz mappings (see Theorem 1.1, 1.2 and Corollaries). In the second part we introduce the new concept o/ s-quasibounded mappings de]ined on Banach spaces and obtain new s~r]ectivity theorems extending results previously obta- ined by other Authors (see Theorem 2.1, 2.2). In the third section, by restricting our atten- tion to the class o] asymptotically linear mappings, we give some extimates o] the s-quasinorm (see Theorem 3.1). As a byproduct we obtain a result of indipendent interest, regarding con- tinuous semigroups (see Corollary 3.4). 1~inally, Theorem 3.3 gives a measure o] sur]ectivity for the class o/strongly asymptotically linear mappings. Introduction. In the last few years the class of a-Lipsehitz mappings (see definition below) has been intensively studied. The survey paper of B. N. SADovsI¢IJ [14] furnishes a brief historical introduction to the subject matter and a comprehensive (though not complete) bibliography. Our aim here is to give some new results regarding surjectivity and differen- tiabflity of ~-Lipschitz mappings. In the first section of the present note we extend to the context of ~-Lipsehitz mappings some well known results regarding Fr6chet and asymptotic derivatives of completely continuous mappings. (See [1, 8, 11, 14]). In the second section we introduce the class of s-quasibounded mappings rep- resenting a natural extension of the concept of quasibounded ones.
    [Show full text]
  • Notes on Functional Analysis
    5. Functional analysis 5.1. Normed spaces and linear maps. For this section, X will denote a vector space over F = R or C.(WewillassumeF = C unless stated otherwise.) Definition 5.1.1. A seminorm on X is a function : X [0, )whichis k·k ! 1 (homogeneous) λx = λ x • (subadditive) xk+ yk | |·x k+k y • k kk k k k We call a norm if in addition it is k·k (definite) x =0impliesx =0. • k k Recall that given a norm on a vector space X, d(x, y):= x y is a metric which k·k k − k induces the norm topology on X.Twonorms 1, 2 are called equivalent if there is a c>0suchthat k·k k·k 1 c− x x c x x X. k k1 k k2 k k1 8 2 Exercise 5.1.2. Show that all norms on Fn are equivalent. Deduce that a finite dimensional subspace of a normed space is closed. Note: You may assume that the unit ball of Fn is compact in the Euclidean topology. Exercise 5.1.3. Show that two norms 1, 2 on X are equivalent if and only if they induce the same topology. k·k k·k Definition 5.1.4. A Banach space is a normed vector space which is complete in the induced metric topology. Examples 5.1.5. (1) If X is an LCH topological space, then C0(X)andCb(X)areBanachspaces. (2) If (X, ,µ)isameasurespace, 1(X, ,µ)isaBanachspace. 1 M L M (3) ` := (xn) F1 xn < { ⇢ | | | 1} Definition 5.1.6.
    [Show full text]
  • Topics in Analysis Part III, Autumn 2010
    Topics in Analysis Part III, Autumn 2010 T. W. K¨orner April 23, 2016 Small print This is just a first draft for the course. The content of the course will be what I say, not what these notes say. Experience shows that skeleton notes (at least when I write them) are very error prone so use these notes with care. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. Contents 1 Introduction 2 2 Baire category 3 3 Non-existence of functions of several variables 5 4 The principle of uniform boundedness 7 5 Countable choice and Baire’s theorem 11 6 Zorn’s lemma and Tychonov’s theorem 13 7 The Hahn-Banach theorem 18 8 Three more uses of Hahn-Banach 21 9 The Rivlin-Shapiro formula 25 10 Uniqueness of Fourier series 27 11 A first look at distributions 31 1 12 The support of distribution 35 13 Distributions on T 37 1 Introduction This course splits into two parts. The first part takes a look at the Baire category theorem, Tychonov’s theorem the Hahn Banach theorem together with some of their consequences. There will be two or three lectures of fairly abstract set theory but the the rest of the course is pretty concrete. The second half of the course will look at the theory of distributions. (The general approach is that of [3] but the main application is different.) I shall therefore assume that you know what is a normed space, and what is a a linear map and that you can do the following exercise.
    [Show full text]
  • Developments in Mathematics
    Developments in Mathematics VOLUME 24 Series Editors: Krishnaswami Alladi, University of Florida Hershel M. Farkas, Hebrew University of Jerusalem Robert Guralnick, University of Southern California For further volumes: www.springer.com/series/5834 Jerzy Kakol ˛ Wiesław Kubis´ Manuel López-Pellicer Descriptive Topology in Selected Topics of Functional Analysis Jerzy Kakol ˛ Wiesław Kubis´ Faculty of Mathematics and Informatics Institute of Mathematics A. Mickiewicz University Jan Kochanowski University 61-614 Poznan 25-406 Kielce Poland Poland [email protected] and Institute of Mathematics Manuel López-Pellicer Academy of Sciences of the Czech Republic IUMPA 115 67 Praha 1 Universitat Poltècnica de València Czech Republic 46022 Valencia [email protected] Spain and Royal Academy of Sciences 28004 Madrid Spain [email protected] ISSN 1389-2177 ISBN 978-1-4614-0528-3 e-ISBN 978-1-4614-0529-0 DOI 10.1007/978-1-4614-0529-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011936698 Mathematics Subject Classification (2010): 46-02, 54-02 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
    [Show full text]
  • The Mackey-Arens and Hahn-Banach Theorems for Spaces Over Valued fields Annales Mathématiques Blaise Pascal, Tome 2, No 1 (1995), P
    ANNALES MATHÉMATIQUES BLAISE PASCAL JERZY K ˛AKOL The Mackey-Arens and Hahn-Banach theorems for spaces over valued fields Annales mathématiques Blaise Pascal, tome 2, no 1 (1995), p. 147-153 <http://www.numdam.org/item?id=AMBP_1995__2_1_147_0> © Annales mathématiques Blaise Pascal, 1995, tous droits réservés. L’accès aux archives de la revue « Annales mathématiques Blaise Pascal » (http:// math.univ-bpclermont.fr/ambp/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction 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/ Ann. Math. Blaise Pascal, Vol. 2, N° 1, 1995, pp.147-153 THE MACKEY-ARENS AND HAHN-BANACH THEOREMS FOR SPACES OVER VALUED FIELDS Jerzy K0105kol Astract. Characterizations of the spherical completeness of a non-archimedean complete non-trivially valued field in terms of classical theorems of Functional Analysis are obtained. 1991 Mathematics subject classification: : ,~651 D Spherical completeness Throughout this paper K = (h’, ( )) will denote a non-archimedean complete valued field with a non-trivial valuation (. I. It is well-known that the absolute value function I . | of the field of the real numbers IR or the complex numbers. C satisfies the following properties : (i) 0 H, |x| = 0 iff x = 0, (ii) |x + |x| + |y|, (iii) |xy| = |x||y|, x, y E IR or x, y ~C. If K is a field, then by a valuation on K we will mean a map | .
    [Show full text]
  • Functional Analysis 7211 Autumn 2017 Homework Problem List
    Functional Analysis 7211 Autumn 2017 Homework problem list Problem 1. Suppose X is a normed space and Y ⊂ X is a subspace. Define Q : X ! X=Y by Qx = x + Y . Define kQxkX=Y = inf fkx − ykX jy 2 Y g : (1) Prove that k · kX=Y is a well-defined seminorm. (2) Show that if Y is closed, then k · kX=Y is a norm. (3) Show that in the case of (2) above, Q : X ! X=Y is continuous and open. Optional: is Q continuous or open only in the case of (1)? (4) Show that if X is Banach, so is X=Y . Problem 2. Suppose F is a finite dimensional vector space. (1) Show that for any two norms k · k1 and k · k2 on F , there is a c > 0 such that kfk1 ≤ ckfk2 for all f 2 F . Deduce that all norms on F induce the same vector space topology on F . (2) Show that for any two finite dimensional normed spaces F1 and F2, all linear maps T : F1 ! F2 are continuous. Optional: Show that for any two finite dimensional vector spaces F1 and F2 endowed with their vector space topologies from part (1), all linear maps T : F1 ! F2 are continuous. (3) Let X; F be normed spaces with F finite dimensional, and let T : X ! F be a linear map. Prove that the following are equivalent: (a) T is bounded, and (b) ker(T ) is closed. Hint: One way to do (b) implies (a) uses Problem 1 part (3) and part (2) of this problem.
    [Show full text]
  • Translation-Invariant Linear Operators
    Mathematical Proceedings of the Cambridge Philosophical Society http://journals.cambridge.org/PSP Additional services for Mathematical Proceedings of the Cambridge Philosophical Society: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here Translation-invariant linear operators H. G. Dales and A. Millinoton Mathematical Proceedings of the Cambridge Philosophical Society / Volume 113 / Issue 01 / January 1993, pp 161 - 172 DOI: 10.1017/S030500410007585X, Published online: 24 October 2008 Link to this article: http://journals.cambridge.org/abstract_S030500410007585X How to cite this article: H. G. Dales and A. Millinoton (1993). Translation-invariant linear operators. Mathematical Proceedings of the Cambridge Philosophical Society, 113, pp 161-172 doi:10.1017/ S030500410007585X Request Permissions : Click here Downloaded from http://journals.cambridge.org/PSP, IP address: 148.88.176.132 on 20 Nov 2013 Math. Proc. Camb. Phil. Soc. (1993). 113, 161 161 Printed in Great Britain Translation-invariant linear operators BY H. G. DALES AND A. MTLLIXGTON School of Mathematics, University of Leeds, Leeds LS2 9JT (Received 14 April 1992) The theory of translation-invariant operators on various spaces of functions (or measures or distributions) is a well-trodden field. The problem is to decide, first, whether or not a linear operator between two function spaces on, say, IR or R+ which commutes with one or many translations on the two spaces is necessarily continuous, and, second, to give a canonical form for all such continuous operators. In some cases each such operator is zero. The second problem is essentially the 'multiplier problem', and it has been extensively discussed; see [7], for example.
    [Show full text]
  • Institute of Mathematics the Czech a Cademy of Sciences
    INSTITUTE OF MATHEMATICS Separable quotients in Cc(X), Cp(X), and their duals Jerzy Kąkol Stephen A. Saxon THE CZECH ACADEMY OF SCIENCES Preprint No. 62-2016 PRAHA 2016 Separable quotients in Cc (X), Cp (X), and their duals Jerzy K ¾akol and Stephen A. Saxon Abstract. The quotient problem has a positive solution for the weak and strong duals of Cc (X) (X an in…nite Tichonov space), for Banach spaces Cc (X) [Rosenthal], and even for barrelled Cc (X), but not for barrelled spaces in general [KST]. The solution is unknown for general Cc (X). A locally convex space is properly separable if it has a proper dense 0-dimensional subspace @ [Robertson]. For Cc (X) quotients, properly separable coincides with in…nite- dimensional separable. Cc (X) has a properly separable algebra quotient if X has a compact denumerable set [Rosenthal]. Relaxing compact to closed, we obtain the converse as well; likewise for Cp (X). And the weak dual of Cp (X), which always has an 0-dimensional quotient, has no properly separable quo- @ tient precisely when X is a P-space. 1. Introduction Here we assume topological vector spaces (tvs’s)and their quotients are Haus- dor¤ and in…nite-dimensional. Banach’sfamous unsolved problem asks: Does every Banach space admit a separable quotient? [Popov’s] Our negative [tvs] lcs solu- tion found a [metrizable tvs] barrelled lcs withouth a separablei quotienth [16,i 20]. (By lcs we mean locally convexh tvs over thei real or complex scalar …eld.) The familiar Banach spaces admit separable quotients, as do all non-Banach Fréchet spaces [6, Satz 2] and (LF )-spaces [28, Theorem 3].
    [Show full text]