Smooth Norms in Orlicz Spaces

1991 ◽  
Vol 34 (1) ◽  
pp. 74-82 ◽  
Author(s):  
R. P. Maleev ◽  
S. L. Troyanski
Keyword(s):  
Orlicz Spaces ◽  
Equivalent Norm ◽  
Equivalent Norms ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Fréchet Differentiable

AbstractEquivalent norms with best order of Frechet and uniformly Frechet differentiability in Orlicz spaces are constructed. Classes of Orlicz which admit infinitely many times Frechet differentiable equivalent norm are found.

A Dual characterization of Banach Spaces With the Convex Point-of-Continuity Property

1989 ◽  
Vol 32 (3) ◽  
pp. 274-280
Author(s):  
D. E. G. Hare
Keyword(s):  
Banach Spaces ◽  
Continuity Property ◽  
Dual Spaces ◽  
Equivalent Norms ◽  
Fréchet Differentiability ◽  
New Type ◽  
Frechet Differentiability ◽  
Point Of Continuity Property ◽  
Convex Point

AbstractWe introduce a new type of differentiability, called cofinite Fréchet differentiability. We show that the convex point-of-continuity property of Banach spaces is dual to the cofinite Fréchet differentiability of all equivalent norms. A corresponding result for dual spaces with the weak* convex point-of-continuity property is also established.

Smoothness, Convexity, Porosity, and Separable Determination

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Equivalent Norm ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Set Of Points

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

scholarly journals Statistical expansions and locally uniform Fréchet differentiability

1991 ◽  
Vol 50 (1) ◽  
pp. 88-97 ◽  
Author(s):  
T. Bednarski ◽  
B. R. Clarke ◽  
W. Kolkiewicz
Keyword(s):  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Fréchet Differentiable ◽  
Uniform Expansions ◽  
Uniform Expansion

AbstractEstimators which have locally uniform expansions are shown in this paper to be asymptotically equivalent to M-estimators. The M-functionals corresponding to these M-estimators are seen to be locally uniformly Fréchet differentiable. Other conditions for M-functionals to be locally uniformly Fréchet differentiable are given. An example of a commonly used estimator which is robust against outliers is given to illustrate that the locally uniform expansion need not be valid.

A Non-Reflexive Smooth Space with a Smooth Dual

1974 ◽  
Vol 17 (4) ◽  
pp. 579-580
Author(s):  
J. H. M. Whitfield
Keyword(s):  
Banach Space ◽  
Real Banach Space ◽  
Gâteaux Differentiability ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Gateaux Differentiability ◽  
Fréchet Differentiable ◽  
Smooth Space ◽  
Nonreflexive Space

Let (E, ρ) and (E*ρ*) be a real Banach space and its dual. Restrepo has shown in [4] that, if p and ρ* are both Fréchet differentiable, E is reflexive. The purpose of this note is to show that Fréchet differentiability cannot be replaced by Gateaux differentiability. This answers negatively a question raised by Wulbert [5]. In particular, we will renorm a certain nonreflexive space with a smooth norm whose dual is also smooth.

scholarly journals Monotonicity and the Dominated Farthest Points Problem in Banach Lattice

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
H. R. Khademzadeh ◽  
H. Mazaheri
Keyword(s):  
Orlicz Function ◽  
Orlicz Spaces ◽  
Farthest Points ◽  
Strong Solvability ◽  
Fréchet Differentiability ◽  
Measure Spaces ◽  
Nonatomic Measure ◽  
Frechet Differentiability ◽  
Farthest Point ◽  
Farthest Point Map

We introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such thatXbecomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spacesL ​ϕ(μ)andE ​ϕ(μ)over nonatomic measure spaces in terms of the functionϕ. We will prove that the Fréchet differentiability of the farthest point map and the conditionsϕ∈Δ2andϕ>0in reflexive Musielak-Orlicz function spaces are equivalent.

Preliminaries to Main Results

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Flat Surface ◽  
Tensor Products ◽  
Linear Operators ◽  
Euclidean Spaces ◽  
Lipschitz Maps ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Fréchet Differentiable ◽  
Derivatives Of ◽  
Surface Passing

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of n-dimensional surfaces. The idea is to deform a flat surface passing through a point x (along which we imagine that a certain function f is almost affine) to a surface passing through a point witnessing that f is not Fréchet differentiable at x. This is done in such a way that certain “energy” associated to surfaces increases less than the “energy” of the functionf along the surface. The chapter also discusses linear operators and tensor products, various notions and notation related to Fréchet differentiability, and deformation of surfaces controlled by ω‎ⁿ. Finally, it proves some integral estimates of derivatives of Lipschitz maps between Euclidean spaces (not necessarily of the same dimension).

Equivalent norms on Banach Jordan algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 261-270 ◽  
Author(s):  
M. A. Youngson
Keyword(s):  
Jordan Algebra ◽  
Sufficient Conditions ◽  
Equivalent Norm ◽  
Jordan Algebras ◽  
The Self ◽  
Equivalent Norms ◽  
Positive Elements ◽  
Definition Of ◽  
Necessary And Sufficient

1. Introduction. Recently Kaplansky suggested the definition of a suitable Jordan analogue of B*-algebras, which we call J B*-algebras (see (10) and (11)). In this article, we give a characterization of those complex unital Banach Jordan algebras which are J B*-algebras in an equivalent norm. This is done by generalizing results of Bonsall ((3) and (4)) to give necessary and sufficient conditions on a real unital Banach Jordan algebra under which it is the self-adjoint part of a J B*-algebra in an equivalent norm. As a corollary we also obtain a characterization of the cones in a Banach Jordan algebra which are the set of positive elements of a J B*-algebra.

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