DUAL DIFFERENTIATION SPACES

2019 ◽  
Vol 99 (03) ◽  
pp. 467-472
Author(s):  
WARREN B. MOORS ◽  
NEŞET ÖZKAN TAN
Keyword(s):  
Banach Space ◽  
Open Problem ◽  
Dense Subset ◽  
Studia Math ◽  
Dual Norm ◽  
Dual Differentiation ◽  
Rotund Norm ◽  
Norm Attaining

We show that if $(X,\Vert \cdot \Vert )$ is a Banach space that admits an equivalent locally uniformly rotund norm and the set of all norm-attaining functionals is residual then the dual norm $\Vert \cdot \Vert ^{\ast }$ on $X^{\ast }$ is Fréchet at the points of a dense subset of $X^{\ast }$ . This answers the main open problem in a paper by Guirao, Montesinos and Zizler [‘Remarks on the set of norm-attaining functionals and differentiability’, Studia Math. 241 (2018), 71–86].

scholarly journals A dual differentiation space without an equivalent locally uniformly rotund norm

2004 ◽  
Vol 77 (3) ◽  
pp. 357-364 ◽  
Author(s):  
Petar S. Kenderov ◽  
Warren B. Moors
Keyword(s):  
Dense Subset ◽  
Continuum Hypothesis ◽  
Open Convex ◽  
Residual Subset ◽  
Continuous Convex Function ◽  
Dual Differentiation ◽  
Fréchet Differentiable ◽  
Rotund Norm ◽  
Open Convex Subset ◽  
The Continuum

AbstractA Banach space (X, ∥ · ∥) is said to be a dual differentiation space if every continuous convex function defined on a non-empty open convex subset A of X* that possesses weak* continuous subgradients at the points of a residual subset of A is Fréchet differentiable on a dense subset of A. In this paper we show that if we assume the continuum hypothesis then there exists a dual differentiation space that does not admit an equivalent locally uniformly rotund norm.

scholarly journals BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

2019 ◽  
pp. 1-17 ◽  
Author(s):  
Johann Langemets ◽  
Ginés López-Pérez
Keyword(s):  
Banach Space ◽  
Studia Math ◽  
Direct Consequence ◽  
Baire Class ◽  
Dual Norm ◽  
Separable Predual ◽  
Separable Case ◽  
Strongly Regular ◽  
First Baire Class

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

scholarly journals A renorming theorem for dual spaces

1983 ◽  
Vol 35 (3) ◽  
pp. 334-337
Author(s):  
A. C. Yorke
Keyword(s):  
Banach Space ◽  
Dense Subset ◽  
Transfinite Induction ◽  
Dual Spaces ◽  
Induction Argument ◽  
Second Dual ◽  
Rotund Norm

AbstractIf the second dual of a Banach space E is smooth at each point of a certain norm dense subset, then its first dual admits a long sequence of norm one projections, and these projections have ranges which are suitable for a transfinite induction argument. This leads to the construction of an equivalent locally uniformly rotund norm and a Markuschevich basis for E*.

scholarly journals On a Characterisation of Differentiability of the Norm of a Normed Linear Space

1971 ◽  
Vol 12 (1) ◽  
pp. 106-114 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Dual Norm ◽  
Uniform Rotundity ◽  
Strong Differentiability ◽  
Differentiability Properties

The purpose of this paper is to show that the various differentiability conditions for the norm of a normed linear space can be characterised by continuity conditions for a certain mapping from the space into its dual. Differentiability properties of the norm are often more easily handled using this characterisation and to demonstrate this we give somewhat more direct proofs of the reflexivity of a Banach space whose dual norm is strongly differentiable, and the duality of uniform rotundity and uniform strong differentiability of the norm for a Banach space.

scholarly journals Uniformly weak differentiability of the norm and a condition of Vlasov

1976 ◽  
Vol 21 (4) ◽  
pp. 393-409 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Smoothness Condition ◽  
Dual Norm ◽  
Weak Differentiability

AbstractIn determining geometrical conditions on a Banach space under which a Chebychev set is convex, Vlasov (1967) introduced a smoothness condition of some interest in itself. Equivalent forms of this condition are derived and it is related to uniformly weak differentiability of the norm and rotundity of the dual norm.

scholarly journals The convex function determined by a multifunction

1996 ◽  
Vol 54 (1) ◽  
pp. 87-97 ◽  
Author(s):  
M. Coodey ◽  
S. Simons
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Convex Function ◽  
Monotone Operator ◽  
Open Problem ◽  
Maximal Monotone Operator ◽  
Convex Functions ◽  
Maximal Monotone ◽  
Minimax Theorem ◽  
Local Boundedness

We shall show how each multifunction on a Banach space determines a convex function that gives a considerable amount of information about the structure of the multifunction. Using standard results on convex functions and a standard minimax theorem, we strengthen known results on the local boundedness of a monotone operator, and the convexity of the interior and closure of the domain of a maximal monotone operator. In addition, we prove that any point surrounded by (in a sense made precise) the convex hull of the domain of a maximal monotone operator is automatically in the interior of the domain, thus settling an open problem.

scholarly journals A NOTE ON L2-SUMMAND VECTORS IN DUAL SPACES

2008 ◽  
Vol 50 (3) ◽  
pp. 429-432 ◽  
Author(s):  
ANTONIO AIZPURU ◽  
FRANCISCO J GARCÍA-PACHECO
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Dual Spaces ◽  
Equivalent Norms ◽  
Norm Attaining

AbstractIt is shown that every L2-summand vector of a dual real Banach space is a norm-attaining functional. As consequences, the L2-summand vectors of a dual real Banach space can be determined by the L2-summand vectors of its predual; for every n ∈ , every real Banach space can be equivalently renormed so that the set of norm-attaining functionals is n-lineable; and it is easy to find equivalent norms on non-reflexive dual real Banach spaces that are not dual norms.

scholarly journals On ranges of variants of the divisor functions that are dense

2015 ◽  
Vol 11 (06) ◽  
pp. 1905-1912 ◽  
Author(s):  
Colin Defant
Keyword(s):  
Real Number ◽  
Open Problem ◽  
Dense Subset ◽  
Arithmetic Function ◽  
Divisor Functions ◽  
Positive Integers ◽  
Multiplicative Arithmetic ◽  
Multiplicative Arithmetic Function

For a real number t, let st be the multiplicative arithmetic function defined by [Formula: see text] for all primes p and positive integers α. We show that the range of a function s-r is dense in the interval (0, 1] whenever r ∈ (0, 1]. We then find a constant ηA ≈ 1.9011618 and show that if r > 1, then the range of the function s-r is a dense subset of the interval [Formula: see text] if and only if r ≤ ηA. We end with an open problem.

scholarly journals Norm Attaining Multilinear Forms on

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Yousef Saleh
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Bilinear Forms ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Multilinear Forms ◽  
Arbitrary Measure ◽  
Norm Attaining Operators ◽  
Norm Attaining

Given an arbitrary measure , this study shows that the set of norm attaining multilinear forms is not dense in the space of all continuous multilinear forms on . However, we have the density if and only if is purely atomic. Furthermore, the study presents an example of a Banach space in which the set of norm attaining operators from into is dense in the space of all bounded linear operators . In contrast, the set of norm attaining bilinear forms on is not dense in the space of continuous bilinear forms on .

scholarly journals ON λ-STRICT IDEALS IN BANACH SPACES

2010 ◽  
Vol 83 (2) ◽  
pp. 231-240 ◽  
Author(s):  
TROND A. ABRAHAMSEN ◽  
OLAV NYGAARD
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Studia Math ◽  
Separable Case ◽  
Baire One ◽  
Ideal Property ◽  
Open Question

AbstractWe define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1 then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.

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