Functional Analysis (Under Construction)
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Vector Measures with Variation in a Banach Function Space
VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE O. BLASCO Departamento de An´alisis Matem´atico, Universidad de Valencia, Doctor Moliner, 50 E-46100 Burjassot (Valencia), Spain E-mail:[email protected] PABLO GREGORI Departament de Matem`atiques, Universitat Jaume I de Castell´o, Campus Riu Sec E-12071 Castell´o de la Plana (Castell´o), Spain E-mail:[email protected] Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the K¨othe-Bochner function space defined as the set of measurable functions f :Ω→ X such that the nonnegative functions fX :Ω→ [0, ∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the p- variation (for E = Lp), Φ-variation (for E = LΦ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation VE (X) is then studied. It is shown, amongother thingsand with some restriction ∗ ∗ of absolute continuity of the norms, that (E(X)) = VE (X ), that VE (X) can be identified with space of cone absolutely summingoperators from E into X and that E(X)=VE (X) if and only if X has the RNP property. 1. Introduction The concept of variation in the frame of vector measures has been fruitful in several areas of the functional analysis, such as the description of the duality of vector-valued function spaces such as certain K¨othe-Bochner function spaces (Gretsky and Uhl10, Dinculeanu7), the reformulation of operator ideals such as the cone absolutely summing operators (Blasco4), and the Hardy spaces of harmonic function (Blasco2,3). -
Uniform Boundedness Principle for Unbounded Operators
UNIFORM BOUNDEDNESS PRINCIPLE FOR UNBOUNDED OPERATORS C. GANESA MOORTHY and CT. RAMASAMY Abstract. A uniform boundedness principle for unbounded operators is derived. A particular case is: Suppose fTigi2I is a family of linear mappings of a Banach space X into a normed space Y such that fTix : i 2 Ig is bounded for each x 2 X; then there exists a dense subset A of the open unit ball in X such that fTix : i 2 I; x 2 Ag is bounded. A closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings as consequences of this principle. Some applications of this principle are also obtained. 1. Introduction There are many forms for uniform boundedness principle. There is no known evidence for this principle for unbounded operators which generalizes classical uniform boundedness principle for bounded operators. The second section presents a uniform boundedness principle for unbounded operators. An application to derive Hellinger-Toeplitz theorem is also obtained in this section. A JJ J I II closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings in the third section as consequences of this principle. Go back Let us assume the following: Every vector space X is over R or C. An α-seminorm (0 < α ≤ 1) is a mapping p: X ! [0; 1) such that p(x + y) ≤ p(x) + p(y), p(ax) ≤ jajαp(x) for all x; y 2 X Full Screen Close Received November 7, 2013. 2010 Mathematics Subject Classification. Primary 46A32, 47L60. Key words and phrases. -
Surjective Isometries on Grassmann Spaces 11
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library SURJECTIVE ISOMETRIES ON GRASSMANN SPACES FERNANDA BOTELHO, JAMES JAMISON, AND LAJOS MOLNAR´ Abstract. Let H be a complex Hilbert space, n a given positive integer and let Pn(H) be the set of all projections on H with rank n. Under the condition dim H ≥ 4n, we describe the surjective isometries of Pn(H) with respect to the gap metric (the metric induced by the operator norm). 1. Introduction The study of isometries of function spaces or operator algebras is an important research area in functional analysis with history dating back to the 1930's. The most classical results in the area are the Banach-Stone theorem describing the surjective linear isometries between Banach spaces of complex- valued continuous functions on compact Hausdorff spaces and its noncommutative extension for general C∗-algebras due to Kadison. This has the surprising and remarkable consequence that the surjective linear isometries between C∗-algebras are closely related to algebra isomorphisms. More precisely, every such surjective linear isometry is a Jordan *-isomorphism followed by multiplication by a fixed unitary element. We point out that in the aforementioned fundamental results the underlying structures are algebras and the isometries are assumed to be linear. However, descriptions of isometries of non-linear structures also play important roles in several areas. We only mention one example which is closely related with the subject matter of this paper. This is the famous Wigner's theorem that describes the structure of quantum mechanical symmetry transformations and plays a fundamental role in the probabilistic aspects of quantum theory. -
Baire Category Theorem and Uniform Boundedness Principle)
Functional Analysis (WS 19/20), Problem Set 3 (Baire Category Theorem and Uniform Boundedness Principle) Baire Category Theorem B1. Let (X; k·kX ) be an innite dimensional Banach space. Prove that X has uncountable Hamel basis. Note: This is Problem A2 from Problem Set 1. B2. Consider subset of bounded sequences 1 A = fx 2 l : only nitely many xk are nonzerog : Can one dene a norm on A so that it becomes a Banach space? Consider the same question with the set of polynomials dened on interval [0; 1]. B3. Prove that the set L2(0; 1) has empty interior as the subset of Banach space L1(0; 1). B4. Let f : [0; 1) ! [0; 1) be a continuous function such that for every x 2 [0; 1), f(kx) ! 0 as k ! 1. Prove that f(x) ! 0 as x ! 1. B5.( Uniform Boundedness Principle) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space. Let fTαgα2A be a family of bounded linear operators between X and Y . Suppose that for any x 2 X, sup kTαxkY < 1: α2A Prove that . supα2A kTαk < 1 Uniform Boundedness Principle U1. Let F be a normed space C[0; 1] with L2(0; 1) norm. Check that the formula 1 Z n 'n(f) = n f(t) dt 0 denes a bounded linear functional on . Verify that for every , F f 2 F supn2N j'n(f)j < 1 but . Why Uniform Boundedness Principle is not satised in this case? supn2N k'nk = 1 U2.( pointwise convergence of operators) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space. -
FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality). -
3 Bilinear Forms & Euclidean/Hermitian Spaces
3 Bilinear Forms & Euclidean/Hermitian Spaces Bilinear forms are a natural generalisation of linear forms and appear in many areas of mathematics. Just as linear algebra can be considered as the study of `degree one' mathematics, bilinear forms arise when we are considering `degree two' (or quadratic) mathematics. For example, an inner product is an example of a bilinear form and it is through inner products that we define the notion of length in n p 2 2 analytic geometry - recall that the length of a vector x 2 R is defined to be x1 + ... + xn and that this formula holds as a consequence of Pythagoras' Theorem. In addition, the `Hessian' matrix that is introduced in multivariable calculus can be considered as defining a bilinear form on tangent spaces and allows us to give well-defined notions of length and angle in tangent spaces to geometric objects. Through considering the properties of this bilinear form we are able to deduce geometric information - for example, the local nature of critical points of a geometric surface. In this final chapter we will give an introduction to arbitrary bilinear forms on K-vector spaces and then specialise to the case K 2 fR, Cg. By restricting our attention to thse number fields we can deduce some particularly nice classification theorems. We will also give an introduction to Euclidean spaces: these are R-vector spaces that are equipped with an inner product and for which we can `do Euclidean geometry', that is, all of the geometric Theorems of Euclid will hold true in any arbitrary Euclidean space. -
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. -
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. -
Sisältö Part I: Topological Vector Spaces 4 1. General Topological
Sisalt¨ o¨ Part I: Topological vector spaces 4 1. General topological vector spaces 4 1.1. Vector space topologies 4 1.2. Neighbourhoods and filters 4 1.3. Finite dimensional topological vector spaces 9 2. Locally convex spaces 10 2.1. Seminorms and semiballs 10 2.2. Continuous linear mappings in locally convex spaces 13 3. Generalizations of theorems known from normed spaces 15 3.1. Separation theorems 15 Hahn and Banach theorem 17 Important consequences 18 Hahn-Banach separation theorem 19 4. Metrizability and completeness (Fr`echet spaces) 19 4.1. Baire's category theorem 19 4.2. Complete topological vector spaces 21 4.3. Metrizable locally convex spaces 22 4.4. Fr´echet spaces 23 4.5. Corollaries of Baire 24 5. Constructions of spaces from each other 25 5.1. Introduction 25 5.2. Subspaces 25 5.3. Factor spaces 26 5.4. Product spaces 27 5.5. Direct sums 27 5.6. The completion 27 5.7. Projektive limits 28 5.8. Inductive locally convex limits 28 5.9. Direct inductive limits 29 6. Bounded sets 29 6.1. Bounded sets 29 6.2. Bounded linear mappings 31 7. Duals, dual pairs and dual topologies 31 7.1. Duals 31 7.2. Dual pairs 32 7.3. Weak topologies in dualities 33 7.4. Polars 36 7.5. Compatible topologies 39 7.6. Polar topologies 40 PART II Distributions 42 1. The idea of distributions 42 1.1. Schwartz test functions and distributions 42 2. Schwartz's test function space 43 1 2.1. The spaces C (Ω) and DK 44 1 2 2.2. -
MAS4107 Linear Algebra 2 Linear Maps And
Introduction Groups and Fields Vector Spaces Subspaces, Linear . Bases and Coordinates MAS4107 Linear Algebra 2 Linear Maps and . Change of Basis Peter Sin More on Linear Maps University of Florida Linear Endomorphisms email: [email protected]fl.edu Quotient Spaces Spaces of Linear . General Prerequisites Direct Sums Minimal polynomial Familiarity with the notion of mathematical proof and some experience in read- Bilinear Forms ing and writing proofs. Familiarity with standard mathematical notation such as Hermitian Forms summations and notations of set theory. Euclidean and . Self-Adjoint Linear . Linear Algebra Prerequisites Notation Familiarity with the notion of linear independence. Gaussian elimination (reduction by row operations) to solve systems of equations. This is the most important algorithm and it will be assumed and used freely in the classes, for example to find JJ J I II coordinate vectors with respect to basis and to compute the matrix of a linear map, to test for linear dependence, etc. The determinant of a square matrix by cofactors Back and also by row operations. Full Screen Close Quit Introduction 0. Introduction Groups and Fields Vector Spaces These notes include some topics from MAS4105, which you should have seen in one Subspaces, Linear . form or another, but probably presented in a totally different way. They have been Bases and Coordinates written in a terse style, so you should read very slowly and with patience. Please Linear Maps and . feel free to email me with any questions or comments. The notes are in electronic Change of Basis form so sections can be changed very easily to incorporate improvements. -
18.102 Introduction to Functional Analysis Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 108 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 19. Thursday, April 16 I am heading towards the spectral theory of self-adjoint compact operators. This is rather similar to the spectral theory of self-adjoint matrices and has many useful applications. There is a very effective spectral theory of general bounded but self- adjoint operators but I do not expect to have time to do this. There is also a pretty satisfactory spectral theory of non-selfadjoint compact operators, which it is more likely I will get to. There is no satisfactory spectral theory for general non-compact and non-self-adjoint operators as you can easily see from examples (such as the shift operator). In some sense compact operators are ‘small’ and rather like finite rank operators. If you accept this, then you will want to say that an operator such as (19.1) Id −K; K 2 K(H) is ‘big’. We are quite interested in this operator because of spectral theory. To say that λ 2 C is an eigenvalue of K is to say that there is a non-trivial solution of (19.2) Ku − λu = 0 where non-trivial means other than than the solution u = 0 which always exists. If λ =6 0 we can divide by λ and we are looking for solutions of −1 (19.3) (Id −λ K)u = 0 −1 which is just (19.1) for another compact operator, namely λ K: What are properties of Id −K which migh show it to be ‘big? Here are three: Proposition 26. -
Functional Analysis 1 Winter Semester 2013-14
Functional analysis 1 Winter semester 2013-14 1. Topological vector spaces Basic notions. Notation. (a) The symbol F stands for the set of all reals or for the set of all complex numbers. (b) Let (X; τ) be a topological space and x 2 X. An open set G containing x is called neigh- borhood of x. We denote τ(x) = fG 2 τ; x 2 Gg. Definition. Suppose that τ is a topology on a vector space X over F such that • (X; τ) is T1, i.e., fxg is a closed set for every x 2 X, and • the vector space operations are continuous with respect to τ, i.e., +: X × X ! X and ·: F × X ! X are continuous. Under these conditions, τ is said to be a vector topology on X and (X; +; ·; τ) is a topological vector space (TVS). Remark. Let X be a TVS. (a) For every a 2 X the mapping x 7! x + a is a homeomorphism of X onto X. (b) For every λ 2 F n f0g the mapping x 7! λx is a homeomorphism of X onto X. Definition. Let X be a vector space over F. We say that A ⊂ X is • balanced if for every α 2 F, jαj ≤ 1, we have αA ⊂ A, • absorbing if for every x 2 X there exists t 2 R; t > 0; such that x 2 tA, • symmetric if A = −A. Definition. Let X be a TVS and A ⊂ X. We say that A is bounded if for every V 2 τ(0) there exists s > 0 such that for every t > s we have A ⊂ tV .