An almost nowhere Fréchet smooth norm on superreflexive spaces

AbstractIf X is a separable non-reflexive Banach space, then the set NA of all norm-attaining elements of X* is not a w*-Gδ subset of X*. However if the norm of X is locally uniformly rotund, then the set of norm attaining elements of norm one is w*-Gδ. There exist separable spaces such that NA is a norm-Borel set of arbitrarily high class. If X is separable and non-reflexive, there exists an equivalent Gâteaux-smooth norm on X such that the set of all Gâteaux-derivatives is not norm-Borel.

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Smoothness, Convexity, Porosity, and Separable Determination

Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179) ◽

10.23943/princeton/9780691153551.003.0003 ◽

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.

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BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2010-043-6 ◽

2010 ◽

Vol 62 (4) ◽

pp. 827-844 ◽

Cited By ~ 4

Author(s):

Caiheng Ouyang ◽

Quanhua Xu

Keyword(s):

Equivalent Norm ◽

Trigonometric Polynomials ◽

Carleson Measures ◽

Uniformly Convex ◽

Bmo Functions ◽

Smooth Norm ◽

Converse Inequality ◽

Lebesgue Measures ◽

The Relationship ◽

Vector Valued

AbstractThis paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let dA and dm denote Lebesgue measures on the unit disc D and the unit circle 𝕋, respectively. For 1 < q < ∞ and a Banach space B, we prove that there exists a positive constant c such thatholds for all trigonometric polynomials f with coefficients in B if and only if B admits an equivalent norm which is q-uniformly convex, whereThe validity of the converse inequality is equivalent to the existence of an equivalent q-uniformly smooth norm.

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Approximation by functions with bounded derivative on Banach spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700034316 ◽

2004 ◽

Vol 69 (1) ◽

pp. 125-131 ◽

Cited By ~ 10

Author(s):

R. Fry

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Open Subset ◽

Separable Banach Space ◽

Smooth Functions ◽

Continuous Map ◽

Smooth Norm ◽

Uniformly Continuous ◽

Bounded Derivative

Let X be a separable Banach space which admits a C1-smooth norm, and let G ⊂ X be an open subset. Then any real-valued, bounded and uniformly continuous map on G can be uniformly approximated on G by C1-smooth functions with bounded derivative.

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Some topics in differential geometry of normed spaces

Advances in Geometry ◽

10.1515/advgeom-2020-0001 ◽

2020 ◽

Vol 0 (0) ◽

Cited By ~ 1

Author(s):

Vitor Balestro ◽

Horst Martini ◽

Ralph Teixeira

Keyword(s):

Differential Geometry ◽

Minimal Surfaces ◽

Dimensional Space ◽

Three Dimensional ◽

Inner Product ◽

Normed Spaces ◽

Smooth Norm ◽

Bonnet Theorem ◽

Three Dimensional Space

AbstractFor a surface immersed in a three-dimensional space endowed with a smooth norm instead of an inner product, one can define analogous concepts of curvature and metric. With such concepts in mind, various questions immediately appear. The aim of this paper is to propose and answer some of those questions. In this framework we prove several characterizations of minimal surfaces in normed spaces, and respective analogues of some global theorems (e.g., Hadamard-type theorems) are also derived. A result on the curvature of surfaces having constant Minkowskian width is given, and finally we study the ambient metric induced on the surface, proving an extension of the classical Bonnet theorem.

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A Non-Reflexive Smooth Space with a Smooth Dual

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-103-4 ◽

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.

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Structure of 6H–SiC(0001) Surfaces from AB-Initio Calculations

Surface Review and Letters ◽

10.1142/s0218625x98000372 ◽

1998 ◽

Vol 05 (01) ◽

pp. 199-205 ◽

Cited By ~ 4

Author(s):

M. Sabisch ◽

P. Krüger ◽

A. Mazur ◽

J. Pollmann

Keyword(s):

Ab Initio Calculations ◽

Ab Initio ◽

Local Density ◽

Substrate Surface ◽

Basis Sets ◽

Smooth Norm ◽

Optimal Surface ◽

Substrate Surfaces ◽

Gaussian Orbital ◽

Good Agreement

We report the results of ab-initio calculations of structural properties of hexagonal 6H–SiC(0001) surfaces. The calculations have been carried out self-consistently within local density approximation employing supercell geometries, smooth norm-conserving pseudopotentials in separable form and Gaussian orbital basis sets. We have investigated several structural models for adatom-induced [Formula: see text] reconstructions with adsorbed Si or C adatoms or trimers residing in threefold-symmetric T 4 or H 3 positions above Si- or C-terminated substrate surfaces, respectively. In the case of the Si-terminated substrate surface our results favor Si adatoms in T 4 sites as optimal configuration in very good agreement with experimental data. For the C-terminated substrate surface our results indicate that none of the investigated [Formula: see text] adatom or trimer configurations is the optimal surface structure.

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Uniqueness of minimal projections in smooth expanded matrix spaces

Monatshefte für Mathematik ◽

10.1007/s00605-020-01471-y ◽

2020 ◽

Author(s):

Michał Kozdęba

Keyword(s):

Three Dimensional ◽

Finite Sets ◽

Minimal Projection ◽

Complex Matrices ◽

New Methods ◽

Smooth Norm ◽

Minimal Projections ◽

Matrix Spaces

Abstract Let us consider the space M(n, m) of all real or complex matrices on n rows and m columns. In 2000 Lesław Skrzypek proved the uniqueness of minimal projection of this space onto its subspace $$M(n,1)+M(1,m)$$ M ( n , 1 ) + M ( 1 , m ) which consists of all sums of matrices with constant rows and matrices with constant columns. We generalize this result using some new methods proved by Lewicki and Skrzypek (J Approx Theory 148:71–91, 2007). Let S be a space of all functions from $$X\times Y \times Z$$ X × Y × Z into $${\mathbb {R}}$$ R or $${\mathbb {C}}$$ C , where X, Y, Z are finite sets. It could be interpreted as a space of three-dimensional matrices M(n, m, r). Let T be a subspace of S consisting of all sums of functions which depend on one variable. Let S be equipped with a smooth norm $$\Vert .\Vert $$ ‖ . ‖ . We show that there exists the unique minimal projection of S onto its subspace T.

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Fr ´Echet Differentiability of Vector-Valued Functions

Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179) ◽

10.23943/princeton/9780691153551.003.0013 ◽

2012 ◽

Author(s):

Joram Lindenstrauss ◽

David Preiss ◽

Tiˇser Jaroslav

Keyword(s):

Mean Value ◽

Common Point ◽

Lipschitz Functions ◽

Lipschitz Map ◽

Fréchet Differentiability ◽

Smooth Norm ◽

Frechet Differentiability ◽

Vector Valued Functions ◽

Vector Valued ◽

Special Case

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has many points of Fréchet differentiability. In particular, it proves that two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. The chapter first presents the theorem whose assumptions hold for any space X with separable dual, includes the result that real-valued Lipschitz functions on such spaces have points of Fréchet differentiability, and takes into account the corresponding mean value estimate. The chapter then gives the estimate for a “regularity parameter” and reduces the theorem to a special case. Finally, it discusses simplifications of the arguments of the proof of the main result in some special situations.

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