On the Dual Space of HP(X)
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Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces
Axioms 2013, 2, 224-270; doi:10.3390/axioms2020224 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Article Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces Sergey Ajiev School of Mathematical Sciences, Faculty of Science, University of Technology-Sydney, P.O. Box 123, Broadway, NSW 2007, Australia; E-Mail: [email protected] Received: 4 March 2013; in revised form: 12 May 2013 / Accepted: 14 May 2013 / Published: 23 May 2013 Abstract: We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schaffer¨ constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Holder-Lipschitz¨ mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration. -
On the Second Dual of the Space of Continuous Functions
ON THE SECOND DUAL OF THE SPACE OF CONTINUOUS FUNCTIONS BY SAMUEL KAPLAN Introduction. By "the space of continuous functions" we mean the Banach space C of real continuous functions on a compact Hausdorff space X. C, its first dual, and its second dual are not only Banach spaces, but also vector lattices, and this aspect of them plays a central role in our treatment. In §§1-3, we collect the properties of vector lattices which we need in the paper. In §4 we add some properties of the first dual of C—the space of Radon measures on X—denoted by L in the present paper. In §5 we imbed C in its second dual, denoted by M, and then identify the space of all bounded real functions on X with a quotient space of M, in fact with a topological direct summand. Thus each bounded function on X represents an entire class of elements of M. The value of an element/ of M on an element p. of L is denoted as usual by/(p.)- If/lies in C then of course f(u) =p(f) (usually written J fdp), and thus the multiplication between Afand P is an extension of the multiplication between L and C. Now in integration theory, for each pEL, there is a stand- ard "extension" of ffdu to certain of the bounded functions on X (we confine ourselves to bounded functions in this paper), which are consequently called u-integrable. The question arises: how is this "extension" related to the multi- plication between M and L. -
A Novel Dual Approach to Nonlinear Semigroups of Lipschitz Operators
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 1, Pages 409{424 S 0002-9947(04)03635-9 Article electronically published on August 11, 2004 A NOVEL DUAL APPROACH TO NONLINEAR SEMIGROUPS OF LIPSCHITZ OPERATORS JIGEN PENG AND ZONGBEN XU Abstract. Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It con- tains C0-semigroup, nonlinear semigroup of contractions and uniformly k- Lipschitzian semigroup as special cases. In this paper, through developing a series of Lipschitz dual notions, we establish an analysis approach to Lips- chitzian semigroup. It is mainly proved that a (nonlinear) Lipschitzian semi- group can be isometrically embedded into a certain C0-semigroup. As ap- plication results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of C0-semigroup are generalized to Lipschitzian semigroup. 1. Introduction Let X and Y be Banach spaces over the same coefficient field K (= R or C), and let C ⊂ X and D ⊂ Y be their subsets. A mapping T from C into D is called Lipschitz operator if there exists a real constant M>0 such that (1.1) k Tx− Ty k≤ M k x − y k; 8x; y 2 C; where the constant M is commonly referred to as a Lipschitz constant of T . Clearly, if the mapping T is a constant (that is, there exists a vector y0 2 D such that Tx= y0 for all x 2 C), then T is a Lipschitz operator. Let Lip(C; D) denote the space of Lipschitz operators from C into D,andfor every T 2 Lip(C; D)letL(T ) denote the minimum Lipschitz constant of T , i.e., k Tx− Ty k (1.2) L(T )= sup : x;y2C;x=6 y k x − y k Then, it can be shown that the nonnegative functional L(·)isaseminormof Lip(C; D) and hence (Lip(C; D);L(·)) is a seminormed linear space. -
Multilinear Algebra
Appendix A Multilinear Algebra This chapter presents concepts from multilinear algebra based on the basic properties of finite dimensional vector spaces and linear maps. The primary aim of the chapter is to give a concise introduction to alternating tensors which are necessary to define differential forms on manifolds. Many of the stated definitions and propositions can be found in Lee [1], Chaps. 11, 12 and 14. Some definitions and propositions are complemented by short and simple examples. First, in Sect. A.1 dual and bidual vector spaces are discussed. Subsequently, in Sects. A.2–A.4, tensors and alternating tensors together with operations such as the tensor and wedge product are introduced. Lastly, in Sect. A.5, the concepts which are necessary to introduce the wedge product are summarized in eight steps. A.1 The Dual Space Let V be a real vector space of finite dimension dim V = n.Let(e1,...,en) be a basis of V . Then every v ∈ V can be uniquely represented as a linear combination i v = v ei , (A.1) where summation convention over repeated indices is applied. The coefficients vi ∈ R arereferredtoascomponents of the vector v. Throughout the whole chapter, only finite dimensional real vector spaces, typically denoted by V , are treated. When not stated differently, summation convention is applied. Definition A.1 (Dual Space)Thedual space of V is the set of real-valued linear functionals ∗ V := {ω : V → R : ω linear} . (A.2) The elements of the dual space V ∗ are called linear forms on V . © Springer International Publishing Switzerland 2015 123 S.R. -
Fact Sheet Functional Analysis
Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 1986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen. Springer, 2000. Triebel, H.: H¨ohere Analysis. Harri Deutsch, 1980. Dobrowolski, M.: Angewandte Funktionalanalysis, Springer, 2010. 1. Banach- and Hilbert spaces Let V be a real vector space. Normed space: A norm is a mapping k · k : V ! [0; 1), such that: kuk = 0 , u = 0; (definiteness) kαuk = jαj · kuk; α 2 R; u 2 V; (positive scalability) ku + vk ≤ kuk + kvk; u; v 2 V: (triangle inequality) The pairing (V; k · k) is called a normed space. Seminorm: In contrast to a norm there may be elements u 6= 0 such that kuk = 0. It still holds kuk = 0 if u = 0. Comparison of two norms: Two norms k · k1, k · k2 are called equivalent if there is a constant C such that: −1 C kuk1 ≤ kuk2 ≤ Ckuk1; u 2 V: If only one of these inequalities can be fulfilled, e.g. kuk2 ≤ Ckuk1; u 2 V; the norm k · k1 is called stronger than the norm k · k2. k · k2 is called weaker than k · k1. Topology: In every normed space a canonical topology can be defined. A subset U ⊂ V is called open if for every u 2 U there exists a " > 0 such that B"(u) = fv 2 V : ku − vk < "g ⊂ U: Convergence: A sequence vn converges to v w.r.t. the norm k · k if lim kvn − vk = 0: n!1 1 A sequence vn ⊂ V is called Cauchy sequence, if supfkvn − vmk : n; m ≥ kg ! 0 for k ! 1. -
Inner Product Spaces
CHAPTER 6 Woman teaching geometry, from a fourteenth-century edition of Euclid’s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2 and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products. Our standing assumptions are as follows: 6.1 Notation F, V F denotes R or C. V denotes a vector space over F. LEARNING OBJECTIVES FOR THIS CHAPTER Cauchy–Schwarz Inequality Gram–Schmidt Procedure linear functionals on inner product spaces calculating minimum distance to a subspace Linear Algebra Done Right, third edition, by Sheldon Axler 164 CHAPTER 6 Inner Product Spaces 6.A Inner Products and Norms Inner Products To motivate the concept of inner prod- 2 3 x1 , x 2 uct, think of vectors in R and R as x arrows with initial point at the origin. x R2 R3 H L The length of a vector in or is called the norm of x, denoted x . 2 k k Thus for x .x1; x2/ R , we have The length of this vector x is p D2 2 2 x x1 x2 . p 2 2 x1 x2 . k k D C 3 C Similarly, if x .x1; x2; x3/ R , p 2D 2 2 2 then x x1 x2 x3 . k k D C C Even though we cannot draw pictures in higher dimensions, the gener- n n alization to R is obvious: we define the norm of x .x1; : : : ; xn/ R D 2 by p 2 2 x x1 xn : k k D C C The norm is not linear on Rn. -
On the Isoperimetric Problem for the Laplacian with Robin and Wentzell Boundary Conditions
On the Isoperimetric Problem for the Laplacian with Robin and Wentzell Boundary Conditions James B. Kennedy A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy School of Mathematics and Statistics March 2010 Abstract We consider the problem of minimising the eigenvalues of the Laplacian with ∂u Robin boundary conditions ∂ν + αu = 0 and generalised Wentzell boundary condi- tions ∆u+β ∂u +γu = 0 with respect to the domain Ω RN on which the problem ∂ν ⊂ is defined. For the Robin problem, when α > 0 we extend the Faber-Krahn in- equality of Daners [Math. Ann. 335 (2006), 767–785], which states that the ball minimises the first eigenvalue, to prove that the minimiser is unique amongst do- mains of class C2. The method of proof uses a functional of the level sets to estimate the first eigenvalue from below, together with a rearrangement of the ball’s eigenfunction onto the domain Ω and the usual isoperimetric inequality. We then prove that the second eigenvalue attains its minimum only on the disjoint union of two equal balls, and set the proof up so it works for the Robin p-Laplacian. For the higher eigenvalues, we show that it is in general impossible for a minimiser to exist independently of α> 0. When α< 0, we prove that every eigenvalue behaves like α2 as α , provided only that Ω is bounded with C1 − → −∞ boundary. This generalises a result of Lou and Zhu [Pacific J. Math. 214 (2004), 323–334] for the first eigenvalue. For the Wentzell problem, we (re-)prove general operator properties, including for the less-studied case β < 0, where the problem is ill-posed in some sense. -
HEAT FLOW and QUANTITATIVE DIFFERENTIATION Contents 1
HEAT FLOW AND QUANTITATIVE DIFFERENTIATION TUOMAS HYTONEN¨ AND ASSAF NAOR Abstract. For every Banach space (Y; k · kY ) that admits an equivalent uniformly convex norm we prove that there exists c = c(Y ) 2 (0; 1) with the following property. Suppose that n 2 N and that X is an n-dimensional normed space with unit ball BX . Then for every 1-Lipschitz function cn f : BX ! Y and for every " 2 (0; 1=2] there exists a radius r > exp(−1=" ), a point x 2 BX with x + rBX ⊆ BX , and an affine mapping Λ : X ! Y such that kf(y) − Λ(y)kY 6 "r for every y 2 x+rBX . This is an improved bound for a fundamental quantitative differentiation problem that was formulated by Bates, Johnson, Lindenstrauss, Preiss and Schechtman (1999), and consequently it yields a new proof of Bourgain's discretization theorem (1987) for uniformly convex targets. The strategy of our proof is inspired by Bourgain's original approach to the discretization problem, which takes the affine mapping Λ to be the first order Taylor polynomial of a time-t Poisson evolute of an extension of f to all of X and argues that, under appropriate assumptions on f, there must exist a time t 2 (0; 1) at which Λ is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that Λ well-approximates f on a macroscopically large ball, in which case we show that for our argument to work one cannot use the Poisson semigroup. Nevertheless, our strategy does succeed with the Poisson semigroup replaced by the heat semigroup. -
Regularization and Simulation of Constrained Partial Differential
Regularization and Simulation of Constrained Partial Differential Equations vorgelegt von Diplom-Mathematiker Robert Altmann aus Berlin Von der Fakult¨atII - Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat genehmigte Dissertation Promotionsausschuss: Vorsitzende: Prof. Dr. Noemi Kurt Berichter: Prof. Dr. Volker Mehrmann Berichterin: Prof. Dr. Caren Tischendorf Berichter: Prof. Dr. Alexander Ostermann Tag der wissenschaftlichen Aussprache: 29. 05. 2015 Berlin 2015 Contents Zusammenfassung . v Abstract . vii Published Papers. ix 1. Introduction . 1 Part A Preliminaries 5 2. Differential-algebraic Equations (DAEs) . 6 2.1. Index Concepts . 6 2.1.1. Differentiation Index. 7 2.1.2. Further Index Concepts. 8 2.2. High-index DAEs . 8 2.3. Index Reduction Techniques. 9 2.3.1. Index Reduction by Differentiation . 9 2.3.2. Minimal Extension . 9 3. Functional Analytic Tools . 11 3.1. Fundamentals . 11 3.1.1. Dual Operators and Riesz Representation Theorem . 11 3.1.2. Test Functions and Distributions . 12 3.1.3. Sobolev Spaces . 13 3.1.4. Traces . 14 3.1.5. Poincar´eInequality and Negative Norms . 15 3.1.6. Weak Convergence and Compactness . 17 3.2. Bochner Spaces . 17 3.3. Sobolev-Bochner Spaces . 20 3.3.1. Gelfand Triples . 20 3.3.2. Definition and Embeddings . 21 4. Abstract Differential Equations . 23 4.1. Nemytskii Mapping . 23 4.2. Operator ODEs . 24 4.2.1. First-order Equations . 25 4.2.2. Second-order Equations. 26 4.3. Operator DAEs . 27 i 5. Discretization Schemes . 29 5.1. Spatial Discretization . 29 5.1.1. -
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Arxiv:Math/0405137V1 [Math.RT] 7 May 2004 Oino Disbecniuu Ersnain Fsc Group)
Draft: May 4, 2004 LOCALLY ANALYTIC VECTORS IN REPRESENTATIONS OF LOCALLY p-ADIC ANALYTIC GROUPS Matthew Emerton Northwestern University Contents 0. Terminology, notation and conventions 7 1. Non-archimedean functional analysis 10 2. Non-archimedean function theory 27 3. Continuous, analytic, and locally analytic vectors 43 4. Smooth, locally finite, and locally algebraic vectors 71 5. Rings of distributions 81 6. Admissible locally analytic representations 100 7. Representationsofcertainproductgroups 124 References 135 Recent years have seen the emergence of a new branch of representation theory: the theory of representations of locally p-adic analytic groups on locally convex p-adic topological vector spaces (or “locally analytic representation theory”, for short). Examples of such representations are provided by finite dimensional alge- braic representations of p-adic reductive groups, and also by smooth representations of such groups (on p-adic vector spaces). One might call these the “classical” ex- amples of such representations. One of the main interests of the theory (from the point of view of number theory) is that it provides a setting in which one can study p-adic completions of the classical representations [6], or construct “p-adic interpolations” of them (for example, by defining locally analytic analogues of the principal series, as in [20], or by constructing representations via the cohomology of arithmetic quotients of symmetric spaces, as in [9]). Locally analytic representation theory also plays an important role in the analysis of p-adic symmetric spaces; indeed, this analysis provided the original motivation for its development. The first “non-classical” examples in the theory were found by Morita, in his analysis of the p-adic upper half-plane (the p-adic symmetric arXiv:math/0405137v1 [math.RT] 7 May 2004 space attached to GL2(Qp)) [15], and further examples were found by Schneider and Teitelbaum in their analytic investigations of the p-adic symmetric spaces of GLn(Qp) (for arbitrary n) [19]. -
Chapter 6. Duality 6.1
6.1. Examples of Dual Spaces 1 Chapter 6. Duality 6.1. Examples of Dual Spaces Note. Recall that the dual space of a normed linear space X is the space of all bounded linear functionals from X to the scalar field F, originally denoted B(X, F), but more often denoted X∗. In this section we find the duals of the `p spaces for 1 ≤ p < ∞ and Lp for 1 ≤ p < ∞. Our lack of background in Lebesgue measure and integration does not allow us to give totally rigorous proofs for some of these results. In the following result, the verb “is” means that there is a surjective isometry between the two relevant spaces. Theorem 6.1. (a) The dual of c0 (the space of all sequences which converge to 0, with the sup norm) is `1. (b) The dual of `1 is `∞. 1 1 (c) The dual of `p for 1 <p< ∞ is `q where + = 1. p q Note. Our text is a little unclear (in my opinion) on some of the details of the proof of Theorem 6.1. Therefore, we present a proof that follows the technique of Reed and Simon’s Functional Analysis (see pages 73 and 74), but use the notation of Promislow. 6.1. Examples of Dual Spaces 2 1 Proof of Theorem 6.1(a). The dual of c0 is ` . 1 (1) Let f = (f(1),f(2),...) ∈ c0 and let g = (g(1), g(2),...) ∈ ` . Then we claim ∞ the mapping φg(f)= f(k)g(k) is a bounded linear functional on c0. -
The First Eigenvalue for the $P$-Laplacian Operator
Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 6, Issue 3, Article 91, 2005 THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR IDRISSA LY LABORATOIRE DE MATHÉMATIQUES ET APPLICATIONS DE METZ ILE DU SAULCY, 57045 METZ CEDEX 01, FRANCE. [email protected] Received 12 February, 2005; accepted 17 June, 2005 Communicated by S.S. Dragomir ABSTRACT. In this paper, using the Hausdorff topology in the space of open sets under some capacity constraints on geometrical domains we prove the strong continuity with respect to the moving domain of the solutions of a p-Laplacian Dirichlet problem. We are also interested in the minimization of the first eigenvalue of the p-Laplacian with Dirichlet boundary conditions among open sets and quasi open sets of given measure. Key words and phrases: p-Laplacian, Nonlinear eigenvalue problems, Shape optimization. 2000 Mathematics Subject Classification. 35J70, 35P30, 35R35. 1. INTRODUCTION Let Ω be an open subset of a fixed ball D in RN ,N ≥ 2 and 1 < p < +∞. Consider the 1,p ∞ Sobolev space W0 (Ω) which is the closure of C functions compactly supported in Ω for the norm Z Z p p p ||u||1,p = |u(x)| dx + |∇u(x)| dx. Ω Ω The p-Laplacian is the operator defined by 1,p −1,q ∆p : W0 (Ω) −→ W (Ω) p−2 u 7−→ ∆pu = div(|∇u| ∇u), −1,q 1,p 1 1 where W (Ω) is the dual space of W0 (Ω) and we have 1 < p, q < ∞, p + q = 1. We are interested in the nonlinear eigenvalue problem ( p−2 −∆pu − λ|u| u = 0 in Ω, (1.1) u = 0 on ∂Ω.