DOCSLIB.ORG

Vector Seminorms, Spaces with Vector Norm, and Regular Operators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Vector Seminorms, Spaces with Vector Norm, and Regular Operators Dedicated to Professor Philippe G. Ciarlet on his 70th birthday VECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS ROMULUS CRISTESCU The rst section of this paper deals with the properties of some type of vector seminorms. In the second section, vector spaces with vector norm are considered while in the third section some theorems about the regular operators are given. Operators dened on a vector space with vector norm are considered in the fourth section. The terminology is that of [3]. AMS 2000 Subject Classication: 47860. Key words: vector seminorm, regular operators. 1. VECTOR SEMINORMS If X is a vector space over R and Y an ordered vector space, a mapping P of X into Y is said to be a vector seminorm if P (x1 + x2) ≤ P (x1) + P (x2), ∀x1, x2 ∈ X, P (αx) = |α| P (x), ∀α ∈ R, ∀x ∈ X. If X and Y are ordered vector spaces, a vector seminorm P : X → Y is said to be monotone if 0 ≤ a ≤ b ∈ X ⇒ P (a) ≤ P (b) while it is said to be absolute monotone if ± x ≤ c ∈ X ⇒ P (x) ≤ P (c). An ordered vector space is said to be (o)-complete if every majorized subset has a supremum. The following theorem generalize some results given in [7] for real semi- norms. Theorem 1.1. Let X be a directed vector space, G a majorized vector subspace of X, and Y an (o)-complete ordered vector space. If P0 : G → Y is REV. ROUMAINE MATH. PURES APPL., 53 (2008), 56, 407418 408 Romulus Cristescu 2 a vector seminorm, then by putting (1.1) P (x) = inf {P0(a) + P0(z) | a, z ∈ G; ± (x − z) ≤ a} , ∀x ∈ X, the following statements hold. (i) The operator P : X → Y given by (1.1) is an absolute monotone vector seminorm and P (z) ≤ P0(z), ∀z ∈ G. (ii) If Q : X → Y is an absolute monotone vector seminorm and if Q(z) ≤ P0(z), ∀z ∈ G, then Q(x) ≤ P (x), ∀x ∈ X. (iii) We have P | G = P0 if and only if the vector seminorm P0 is absolute monotone. Proof. (i) For any x ∈ X there exists the element P (x) given by (1.1). Indeed, since G is a majorizing vector subspace of the directed vector space X, for any elements x ∈ X and z ∈ G there exists an element a ∈ G such that ± (x − z) ≤ a. On the other hand, we have P0 (c) ≥ 0, ∀c ∈ G, and the space Y is (o)-complete. As in the case of real seminorms, it is easily veried that the operator P given by (1.1) is a vector seminorm. We also have (1.2) P (x) = inf {P0 (b) | x ≤ b ∈ G} , ∀x ∈ X+. Indeed, denoting by P 0(x) the right-hand side in (1.2), we obviously have 0 P (x) ≤ P (x). Let now x ∈ X+ and ± (x − z) ≤ a with a, z ∈ G. Then 0 ≤ x ≤ a + z ∈ G, whence 0 P (x) ≤ P0 (a + z) ≤ P0 (a) + P0(z). So, P 0(x) ≤ P (x). Therefore (1.2) holds. The vector seminorm P is absolute monotone. Indeed, if ± x ≤ v in X then, taking a ∈ G such that v ≤ a, it follows from ± x ≤ a that P (x) ≤ P0(a) by (1.1). Then, by (1.2), we get P (x) ≤ P (v). Obviously P (z) ≤ P0(z), ∀z ∈ G. (ii) Let Q : X → Y be an absolute monotone vector seminorm such that Q(z) ≤ P0(z), ∀z ∈ G. If x ∈ X and ± (x − z) ≤ a with a, z ∈ G, then Q(x) ≤ Q(x − z) + Q(z) ≤ Q(a) + Q(z) ≤ P0(a) + P0(z) whence Q(x) ≤ P (x). (iii) Suppose that P0 is absolute monotone and let x ∈ G. Let also, z, a ∈ G such that ± (x − z) ≤ a. We have P0(x − z) ≤ P0(a) whence, as in the preceding case regarding the vector seminorm Q, we obtain P0(x) ≤ P (x). We also have P (x) ≤ P0(x) by statement (i). Conversely, if then is obviously absolute monotone. P | G = P0 P0 3 Vector seminorms, spaces with vector norm, and regular operators 409 Denition 1.1. If X is a directed vector space and Y an (o)-complete ordered vector space, a vector seminorm P : X → Y is said to be solid if P (x) = inf {P (a) | ± x ≤ a ∈ X} , ∀x ∈ X. Remark 1.1. Let X be a directed vector space and Y an (o)-complete ordered vector space. If G is a majorizing vector subspace of X and P0 : G → Y a solid vector seminorm, then the formula P (x) = inf{P0(a) | ± x ≤ a ∈ G}, ∀x ∈ X, yields a solid vector seminorm P : X → Y such that P | G = P0. Remark 1.2. Let X be a vector lattice and Y an (o)-complete ordered vector space. A vector seminorm P : X → Y is solid if and only if |x1| ≤ |x2| in X implies P (x1) ≤ P (x2). This assertion is veried as in the case Y = R. Remark 1.3. Let X be a vector lattice, Y an (o)-complete ordered vector space and P : X → Y a solid vector seminorm. Then P is additive on the positive cone X+, if and only if P is of the form (1.3) P (x) = U(|x|), ∀x ∈ X, where U : X → Y is a positive linear operator. Indeed, if P is additive on X+ then, by putting, U(x) = P (x+) − P (x−), ∀x ∈ X, we obtain a positive linear operator U : X → Y for which (1.3) holds. Con- versely, if U : X → Y is a positive linear operator, then by (1.3) we obviously obtain a solid vector seminorm. We recall now that if X and Y are vector lattices, an operator U : X → Y is said [3] to be disjunctive if x1 ⊥ x2 in X implies U(x1) ⊥ U(x2). Theorem 1.2. If X and Y are vector lattices and P : X → Y is a monotone and disjunctive vector seminorm, then (1.4) P (x1 ∧ x2) = P (x1) ∧ P (x2), ∀x1, x2 ∈ X+, and if x1 ∧ x2 = 0 in X, then (1.5) P (x1 ∨ x2) = P (x1) ∨ P (x2). Proof. Let x1, x2 ∈ X+. We have (x1 − (x1 ∧ x2)) ∧ (x2 − (x1 ∧ x2)) = 0, whence (1.6) P (x1 − (x1 ∧ x2)) ∧ P (x2 − (x1 ∧ x2)) = 0. 410 Romulus Cristescu 4 On the other hand 0 ≤ P (xi) − P (x1 ∧ x2) ≤ P (xi − (x1 ∧ x2)), i = 1, 2, and, by (1.6), (P (x1) − P (x1 ∧ x2)) ∧ (P (x2) − P (x1 ∧ x2)) = 0. Therefore, (1.4) holds. If x1 ∧ x2 = 0 then x1 ∨ x2 = x1 + x2, hence P (x1 ∨ x2) = P (x1 + x2) ≤ P (x1) + P (x2) = P (x1) ∨ P (x2) since P (x1) ⊥ P (x2). On the other hand, we have P (x1) ∨ P (x2) ≤ P (x1 ∨ x2), so that (1.5) holds. Remark 1.4. Let X and Y be vector lattices and let P : X → Y be a solid vector seminorm. Then P is disjunctive if and only if x1 ∧ x2 = 0 in X implies P (x1) ∧ P (x2) = 0. Theorem 1.3. Let X be a vector lattice and Y an (o)-complete ordered vector space. If P : X → Y is a (non-zero) solid vector seminorm, then there exists a (non-zero) positive linear operator U : X → Y such that (1.7) |U(x)| ≤ P (x), ∀x ∈ X. Proof. The proof is similar to that given in the case of functionals [4]. By putting Q(x) = P (x+), ∀x ∈ X, we obtain a sublinear operator Q : X → Y . Let us consider an element a > 0 of X such that P (a) > 0. By a corollary of the Hahn-Banach-Kantorovich theorem [5] there exists a linear operator U : X → Y such that U(a) = Q(a) and U(x) ≤ Q(x), ∀x ∈ X. It is easy to see that U is positive and that inequality (1.7) holds. Theorem 1.4. Let X be a vector lattice, Y an (o)-complete ordered vector space and P (X, Y ) the set of all solid vector seminorms mapping X into Y . If Pi ∈ P(X, Y ), i = 1, 2, then, with respect to the pointwise order in the set P(X, Y ), there exists P1 ∧ P2 in this set, and we have (P1 ∧ P2)(x) = inf {P1(a) + P2(b) | a, b ∈ X+; a + b = |x|} , ∀x ∈ X. The proof is similar to that given in the case Y = R (see [2]). 5 Vector seminorms, spaces with vector norm, and regular operators 411 2. SPACES WITH VECTOR NORM Let F be a real vector space and X a vector lattice. A vector seminorm P : F → X is said to be a vector norm if P (x) = 0 implies x = 0. If the vector space F is endowed with a vector norm P : F → X, then F is called a v-normed vector space and is denoted by . We shall also write FX kfkX = P (f). Let FX be a v-normed vector space. A subset A of FX is said to be (v)-bounded if there exists x0 ∈ X such that kfkX ≤ x0, ∀f ∈ A. A sequence of elements of the space is said to be a - (fn)n∈N FX (v) Cauchy sequence if there exists a sequence of elements of such that (an)n∈N X an ↓ 0 and n∈N (2.1) kfi − fjkX ≤ an, ∀i, j ≥ n. We say that a sequence -converges to an element (fn)n∈N (v) f ∈ FX if the sequence (kfn − fk ) (o)-converges (i.e.
Load more

Recommended publications
  • Functional Analysis Lecture Notes Chapter 3. Banach
    FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples Notation 1.1. Throughout, F will denote either the real line R or the complex plane C. All vector spaces are assumed to be over the field F. Definition 1.2. Let X be a vector space over the field F. Then a semi-norm on X is a function k · k: X ! R such that (a) kxk ≥ 0 for all x 2 X, (b) kαxk = jαj kxk for all x 2 X and α 2 F, (c) Triangle Inequality: kx + yk ≤ kxk + kyk for all x, y 2 X. A norm on X is a semi-norm which also satisfies: (d) kxk = 0 =) x = 0. A vector space X together with a norm k · k is called a normed linear space, a normed vector space, or simply a normed space. Definition 1.3. Let I be a finite or countable index set (for example, I = f1; : : : ; Ng if finite, or I = N or Z if infinite). Let w : I ! [0; 1). Given a sequence of scalars x = (xi)i2I , set 1=p jx jp w(i)p ; 0 < p < 1; 8 i kxkp;w = > Xi2I <> sup jxij w(i); p = 1; i2I > where these quantities could be infinite.:> Then we set p `w(I) = x = (xi)i2I : kxkp < 1 : n o p p p We call `w(I) a weighted ` space, and often denote it just by `w (especially if I = N). If p p w(i) = 1 for all i, then we simply call this space ` (I) or ` and write k · kp instead of k · kp;w.
    [Show full text]
  • Elements of Linear Algebra, Topology, and Calculus
    00˙AMS September 23, 2007 © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Appendix A Elements of Linear Algebra, Topology, and Calculus A.1 LINEAR ALGEBRA We follow the usual conventions of matrix computations. Rn×p is the set of all n p real matrices (m rows and p columns). Rn is the set Rn×1 of column vectors× with n real entries. A(i, j) denotes the i, j entry (ith row, jth column) of the matrix A. Given A Rm×n and B Rn×p, the matrix product Rm×p ∈ n ∈ AB is defined by (AB)(i, j) = k=1 A(i, k)B(k, j), i = 1, . , m, j = 1∈, . , p. AT is the transpose of the matrix A:(AT )(i, j) = A(j, i). The entries A(i, i) form the diagonal of A. A Pmatrix is square if is has the same number of rows and columns. When A and B are square matrices of the same dimension, [A, B] = AB BA is termed the commutator of A and B. A matrix A is symmetric if −AT = A and skew-symmetric if AT = A. The commutator of two symmetric matrices or two skew-symmetric matrices− is symmetric, and the commutator of a symmetric and a skew-symmetric matrix is skew-symmetric. The trace of A is the sum of the diagonal elements of A, min(n,p ) tr(A) = A(i, i). i=1 X We have the following properties (assuming that A and B have adequate dimensions) tr(A) = tr(AT ), (A.1a) tr(AB) = tr(BA), (A.1b) tr([A, B]) = 0, (A.1c) tr(B) = 0 if BT = B, (A.1d) − tr(AB) = 0 if AT = A and BT = B.
    [Show full text]
  • Linear Spaces
    Chapter 2 Linear Spaces Contents FieldofScalars ........................................ ............. 2.2 VectorSpaces ........................................ .............. 2.3 Subspaces .......................................... .............. 2.5 Sumofsubsets........................................ .............. 2.5 Linearcombinations..................................... .............. 2.6 Linearindependence................................... ................ 2.7 BasisandDimension ..................................... ............. 2.7 Convexity ............................................ ............ 2.8 Normedlinearspaces ................................... ............... 2.9 The `p and Lp spaces ............................................. 2.10 Topologicalconcepts ................................... ............... 2.12 Opensets ............................................ ............ 2.13 Closedsets........................................... ............. 2.14 Boundedsets......................................... .............. 2.15 Convergence of sequences . ................... 2.16 Series .............................................. ............ 2.17 Cauchysequences .................................... ................ 2.18 Banachspaces....................................... ............... 2.19 Completesubsets ....................................... ............. 2.19 Transformations ...................................... ............... 2.21 Lineartransformations.................................. ................ 2.21
    [Show full text]
  • Basic Differentiable Calculus Review
    Basic Differentiable Calculus Review Jimmie Lawson Department of Mathematics Louisiana State University Spring, 2003 1 Introduction Basic facts about the multivariable differentiable calculus are needed as back- ground for differentiable geometry and its applications. The purpose of these notes is to recall some of these facts. We state the results in the general context of Banach spaces, although we are specifically concerned with the finite-dimensional setting, specifically that of Rn. Let U be an open set of Rn. A function f : U ! R is said to be a Cr map for 0 ≤ r ≤ 1 if all partial derivatives up through order r exist for all points of U and are continuous. In the extreme cases C0 means that f is continuous and C1 means that all partials of all orders exists and are continuous on m r r U. A function f : U ! R is a C map if fi := πif is C for i = 1; : : : ; m, m th where πi : R ! R is the i projection map defined by πi(x1; : : : ; xm) = xi. It is a standard result that mixed partials of degree less than or equal to r and of the same type up to interchanges of order are equal for a C r-function (sometimes called Clairaut's Theorem). We can consider a category with objects nonempty open subsets of Rn for various n and morphisms Cr-maps. This is indeed a category, since the composition of Cr maps is again a Cr map. 1 2 Normed Spaces and Bounded Linear Oper- ators At the heart of the differential calculus is the notion of a differentiable func- tion.
    [Show full text]
  • Embedding Banach Spaces
    Embedding Banach spaces. Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, and Andras Zsak January 24, 2009 Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Definition (Banach space) A normed vector space (X ; k · k) is called a Banach space if X is complete in the norm topology. Definition (Schauder basis) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 unique sequence of scalars (ai )i=1 such that x = i=1 ai xi . Question (Mazur '36) Does every separable Banach space have a basis? Theorem (Enflo '72) No. Background Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Definition (Schauder basis) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 unique sequence of scalars (ai )i=1 such that x = i=1 ai xi . Question (Mazur '36) Does every separable Banach space have a basis? Theorem (Enflo '72) No. Background Definition (Banach space) A normed vector space (X ; k · k) is called a Banach space if X is complete in the norm topology. Daniel Freeman with Edward Odell, Thomas Schlumprecht, Richard Haydon, andEmbedding Andras Zsak Banach spaces. Question (Mazur '36) Does every separable Banach space have a basis? Theorem (Enflo '72) No. Background Definition (Banach space) A normed vector space (X ; k · k) is called a Banach space if X is complete in the norm topology. Definition (Schauder basis) 1 A sequence (xi )i=1 ⊂ X is called a basis for X if for every x 2 X there exists a 1 P1 unique sequence of scalars (ai )i=1 such that x = i=1 ai xi .
    [Show full text]
  • Nonlinear Spectral Theory Henning Wunderlich
    Nonlinear Spectral Theory Henning Wunderlich Dr. Henning Wunderlich, Frankfurt, Germany E-mail address: [email protected] 1991 Mathematics Subject Classification. 46-XX,47-XX I would like to sincerely thank Prof. Dr. Delio Mugnolo for his valuable advice and support during the preparation of this thesis. Contents Introduction 5 Chapter 1. Spaces 7 1. Topological Spaces 7 2. Uniform Spaces 11 3. Metric Spaces 14 4. Vector Spaces 15 5. Topological Vector Spaces 18 6. Locally-Convex Spaces 21 7. Bornological Spaces 23 8. Barreled Spaces 23 9. Metric Vector Spaces 29 10. Normed Vector Spaces 30 11. Inner-Product Vector Spaces 31 12. Examples 31 Chapter 2. Fixed Points 39 1. Schauder-Tychonoff 39 2. Monotonic Operators 43 3. Dugundji and Quasi-Extensions 51 4. Measures of Noncompactness 53 5. Michael Selection 58 Chapter 3. Existing Spectra 59 1. Linear Spectrum and Properties 59 2. Spectra Under Consideration 63 3. Restriction to Linear Operators 76 4. Nonemptyness 77 5. Closedness 78 6. Boundedness 82 7. Upper Semicontinuity 84 Chapter 4. Applications 87 1. Nemyckii Operator 87 2. p-Laplace Operator 88 3. Navier-Stokes Equations 92 Bibliography 97 Index 101 3 Introduction The term Spectral Theory was coined by David Hilbert in his studies of qua- dratic forms in infinitely-many variables. This theory evolved into a beautiful blend of Linear Algebra and Analysis, with striking applications in different fields of sci- ence. For example, the formulation of the calculus of Quantum Mechanics (e.g., POVM) would not have been possible without such a (Linear) Spectral Theory at hand.
    [Show full text]
  • Lp Spaces in Isabelle
    Lp spaces in Isabelle Sebastien Gouezel Abstract Lp is the space of functions whose p-th power is integrable. It is one of the most fundamental Banach spaces that is used in analysis and probability. We develop a framework for function spaces, and then im- plement the Lp spaces in this framework using the existing integration theory in Isabelle/HOL. Our development contains most fundamental properties of Lp spaces, notably the Hölder and Minkowski inequalities, completeness of Lp, duality, stability under almost sure convergence, multiplication of functions in Lp and Lq, stability under conditional expectation. Contents 1 Functions as a real vector space 2 2 Quasinorms on function spaces 4 2.1 Definition of quasinorms ..................... 5 2.2 The space and the zero space of a quasinorm ......... 7 2.3 An example: the ambient norm in a normed vector space .. 9 2.4 An example: the space of bounded continuous functions from a topological space to a normed real vector space ....... 10 2.5 Continuous inclusions between functional spaces ....... 10 2.6 The intersection and the sum of two functional spaces .... 12 2.7 Topology ............................. 14 3 Conjugate exponents 16 4 Convexity inequalities and integration 18 5 Lp spaces 19 5.1 L1 ................................. 20 5.2 Lp for 0 < p < 1 ......................... 21 5.3 Specialization to L1 ....................... 22 5.4 L0 ................................. 23 5.5 Basic results on Lp for general p ................ 24 5.6 Lp versions of the main theorems in integration theory .... 25 1 5.7 Completeness of Lp ........................ 26 5.8 Multiplication of functions, duality ............... 26 5.9 Conditional expectations and Lp ...............
    [Show full text]
  • 1. Homework 7 Let X,Y,Z Be Normed Vector Spaces Over R. (1) Let Bil(X ×Y,Z) Be the Space of Bounded Bilinear Maps from X ×Y Into Z
    1. Homework 7 Let X; Y; Z be normed vector spaces over R: (1) Let bil(X ×Y; Z) be the space of bounded bilinear maps from X ×Y into Z: (For the definition of bounded bilinear maps, see class notes.) For each T 2 bil(X × Y; Z); we define kT k = sup kT (x; y)kZ : kxkX =kykY =1 Prove that (bil(X × Y; Z); k · k) forms a normed vector space. (2) A linear map φ : X ! Y is called an isomorphism of normed vector spaces over R if φ is an isomorphism of vector spaces such that kφ(x)kY = kxkX for any x 2 X: Prove that the map b b ' : L (X; L (Y; Z)) ! bil(X × Y; Z);T 7! 'T defined by 'T (x; y) = T (x)(y) is an isomorphism of normed vector spaces. (3) Compute Df(0; 0);D2f(0; 0) and D3f(0; 0) for the following given functions f: (a) f(x; y) = x4 + y4 − x2 − y2 + 1: (b) f(x; y) = cos(x + 2y): (c) f(x; y) = ex+y: (4) Let f :[a; b] ! R be continuous. Suppose that Z b f(x)h(x)dx = 0 a for any continuous function h :[a; b] ! R with h(a) = h(b) = 0: Show that f is the zero function on [a; b]: (5) Let C1[a; b] be the space of all real valued continuously differentiable functions on [a; b]: On C1[a; b]; we define 0 1 kfkC1 = kfk1 + kf k1 for any f 2 C [a; b]: 1 (a) Show that (C [a; b]; k · kC1 ) is a Banach space over R: 1 (b) Let X be the subset of C [a; b] consisting of functions f :[a; b] ! R so that f(a) = f(b) = 0: Prove that X forms a closed vector subspace of C1[a; b]: Hence X is also a Banach space.
    [Show full text]
  • General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces
    S S symmetry Article General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces Silvestru Sever Dragomir 1,2 1 Mathematics, College of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne 8001, Australia; [email protected] 2 DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa Abstract: In this paper we establish some error bounds in approximating the integral by general trapezoid type rules for Fréchet differentiable functions with values in Banach spaces. Keywords: Banach spaces; norm inequalities; midpoint inequalities; Banach algebras 1. Introduction We recall some facts about differentiation of functions between normed vector spaces [1]. Let O be an open subset of a normed vector space E, f a real-valued function defined on O, a 2 O and u a nonzero element of E. The function fu given by t 7! f (a + tu) is d fu defined on an open interval containing 0. If the derivative dt (0) is defined, i.e., if the limit Citation: Dragomir, S.S. General Norm Inequalities of Trapezoid Type d f f (a + tu) − f (a) u (0) := lim for Fréchet Differentiable Functions dt t!0 t in Banach Spaces. Symmetry 2021, 13, 1288. https://doi.org/10.3390/ exists, then we denote this derivative by ra f (u). It is called the Gâteaux derivative (di- sym13071288 rectional derivative) of f at a in the direction u. If ra f (u) is defined and l 2 Rnf0g, then ra f (lu) is defined and ra f (lu) = lra f (u).
    [Show full text]
  • Normed Vector Spaces
    Chapter II Elements of Functional Analysis Functional Analysis is generally understood a “linear algebra for infinite di- mensional vector spaces.” Most of the vector spaces that are used are spaces of (various types of) functions, therfeore the name “functional.” This chapter in- troduces the reader to some very basic results in Functional Analysis. The more motivated reader is suggested to continue with more in depth texts. 1. Normed vector spaces Definition. Let K be one of the fields R or C, and let X be a K-vector space. A norm on X is a map X 3 x 7−→ kxk ∈ [0, ∞) with the following properties (i) kx + yk ≤ kxk + kyk, ∀ x, y ∈ X; (ii) kλxk = |λ| · kxk, ∀ x ∈ X, λ ∈ K; (iii) kxk = 0 =⇒ x = 0. (Note that conditions (i) and (ii) state that k . k is a seminorm.) Examples 1.1. Let K be either R or C, and let J be some non-empty set. A. Define `∞(J) = α : J → : sup |α(j)| < ∞ . K K j∈J We equip the space `∞(J) with the -vector space structure defined by point-wise K K addition and point-wise scalar multiplication. This vector space carries a natural norm k . k∞, defined by kαk = sup |α(j)|, α ∈ `∞(I). ∞ K j∈J When = , the space `∞(J) is simply denoted by `∞(J). When J = - the set K C C N of natural numbers - instead of `∞( ) we simply write `∞, and instead of `∞( ) R N R N we simply write `∞. B. Consider the space K c0 (J) = α : J → K : inf sup |α(j)| = 0 .
    [Show full text]
  • Notes for Functional Analysis
    Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) November 13, 2015 1 Lecture 18 1.1 Characterizations of reflexive spaces Recall that a Banach space X is reflexive if the inclusion X ⊂ X∗∗ is a Banach space isomorphism. The following theorem of Kakatani give us a very useful creteria of reflexive spaces: Theorem 1.1. A Banach space X is reflexive iff the closed unit ball B = fx 2 X : kxk ≤ 1g is weakly compact. Proof. First assume X is reflexive. Then B is the closed unit ball in X∗∗. By the Banach- Alaoglu theorem, B is compact with respect to the weak-∗ topology of X∗∗. But by definition, in this case the weak-∗ topology on X∗∗ coincides with the weak topology on X (since both are the weakest topology on X = X∗∗ making elements in X∗ continuous). So B is compact with respect to the weak topology on X. Conversly suppose B is weakly compact. Let ι : X,! X∗∗ be the canonical inclusion map that sends x to evx. Then we've seen that ι preserves the norm, and thus is continuous (with respect to the norm topologies). By PSet 8-1 Problem 3, ∗∗ ι : X(weak) ,! X(weak) is continuous. Since the weak-∗ topology on X∗∗ is weaker than the weak topology on X∗∗ (since X∗∗∗ = (X∗)∗∗ ⊃ X∗. OH MY GOD!) we thus conclude that ∗∗ ι : X(weak) ,! X(weak-∗) is continuous. But by assumption B ⊂ X(weak) is compact, so the image ι(B) is compact in ∗∗ X(weak-∗). In particular, ι(B) is weak-∗ closed.
    [Show full text]
  • Vector Spaces
    Vector Spaces Vector space, º, over the ¯eld of complex numbers, C, is a set of elements jai, jbi;:::, satisfying the following axioms. ¦ For each two vectors jai; jbi 2 º there exists a summation procedure: jai + jbi = jci, where jci 2 º. The summation obeys the following laws. jai + jbi = jbi + jai (commutative) ; (1) jai + (jbi + jci) = (jai + jbi) + jci (associative) : (2) ¦ There exists a zero vector j0i, such that 8 jai: jai + j0i = jai : (3) ¦ 8 jai 9 j¡ai (additive inverse) such that jai + j¡ai = j0i : (4) [Here we start using the symbol 8 that means `for all' (`for each', `for any') and the symbol 9 that means `there exists'.] ¦ There exists a procedure of multiplication by a scalar ® 2 C. That is 8 jai 2 º, 8 ® 2 C: 9 ® jai 2 º. Multiplication by a scalar obeys the following laws. ® (¯ jai ) = (®¯) jai ; (5) 1 ¢ jai = jai ; (6) ®(jai + jbi) = ®jai + ®jbi ; (7) (® + ¯)jai = ®jai + ¯jai : (8) From the above axioms it follows that 8 jai 0 ¢ jai = j0i ; (9) (¡1) ¢ j ai = j¡ai : (10) 1 Problem 8. On the basis of the axioms: (a) Show that the zero element is unique. (b) Show that for any vector jai there exists only one additive inverse. (c) Show that for any vector jai the relation jai + jxi = jai implies that jxi = j0i . (d) Derive (9)-(10). Important (!): Here you are not allowed to use the subtraction procedure, which is not de¯ned yet, and cannot be de¯ned prior to establishing the uniqueness of additive inverse.
    [Show full text]