scholarly journals FRÉCHET INTERMEDIATE DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS ON ASPLUND SPACES

2009 ◽  
Vol 79 (2) ◽  
pp. 309-317 ◽  
Author(s):  
J. R. GILES
Keyword(s):  
Large Class ◽  
Open Subset ◽  
Lipschitz Function ◽  
Dense Subset ◽  
Lipschitz Functions ◽  
Nonempty Open Subset ◽  
Asplund Space ◽  
Asplund Spaces ◽  
Fréchet Differentiable

AbstractThe deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

scholarly journals On the characterisation of Asplund spaces

1982 ◽  
Vol 32 (1) ◽  
pp. 134-144 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Direct Proof ◽  
Asplund Space ◽  
Open Convex ◽  
Asplund Spaces ◽  
Separable Subspace ◽  
Nikodym Property ◽  
Continuous Convex Function ◽  
Fréchet Differentiable ◽  
Open Convex Subset

AbstractA Banach space is an Asplund space if every continuous convex function on an open convex subset is Fréchet differentiable on a dense G8 subset of its domain. The recent research on the Radon-Nikodým property in Banach spaces has revealed that a Banach space is an Asplund space if and only if every separable subspace has separable dual. It would appear that there is a case for providing a more direct proof of this characterisation.

On the Composition of Differentiable Functions

2003 ◽  
Vol 46 (4) ◽  
pp. 481-494 ◽  
Author(s):  
M. Bachir ◽  
G. Lancien
Keyword(s):  
Banach Space ◽  
Variational Principle ◽  
Linear Operator ◽  
Differentiable Function ◽  
General Result ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Schur Property ◽  
Differentiable Functions ◽  
Fréchet Differentiable

AbstractWe prove that a Banach space X has the Schur property if and only if every X-valued weakly differentiable function is Fréchet differentiable. We give a general result on the Fréchet differentiability of f ○ T, where f is a Lipschitz function and T is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm ‖ ‖lip on various spaces of Lipschitz functions.

scholarly journals The differentiability of convex functions on topological linear spaces

1990 ◽  
Vol 42 (2) ◽  
pp. 201-213 ◽  
Author(s):  
Bernice Sharp
Keyword(s):  
Convex Functions ◽  
Inductive Limit ◽  
Asplund Space ◽  
Fréchet Spaces ◽  
Linear Spaces ◽  
Asplund Spaces ◽  
Continuous Convex Function ◽  
Mazur’S Theorem ◽  
Topological Linear ◽  
Differentiability Properties

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

scholarly journals An answer to Furstenberg’s problem on topological disjointness

2019 ◽  
Vol 40 (9) ◽  
pp. 2467-2481 ◽  
Author(s):  
WEN HUANG ◽  
SONG SHAO ◽  
XIANGDONG YE
Keyword(s):  
Open Subset ◽  
Dense Subset ◽  
Minimal System ◽  
Open Neighbourhood ◽  
Countable Dense Subset ◽  
Weakly Mixing ◽  
Transitive System

In this paper we give an answer to Furstenberg’s problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\in Y$ and any open neighbourhood $V$ of $y$, and for any non-empty open subset $U\subset X$, there is $x\in D\cap U$ such that $\{n\in \mathbb{Z}_{+}:T^{n}x\in U,S^{n}y\in V\}$ is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^{n},T^{(n)})$ and $(X,T^{n})$ for any $n\in \mathbb{N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_{K})$ is disjoint from all minimal systems.

Random walks on discrete and continuous circles

1993 ◽  
Vol 30 (4) ◽  
pp. 780-789 ◽  
Author(s):  
Jeffrey S. Rosenthal
Keyword(s):  
Random Walk ◽  
Fourier Analysis ◽  
Random Walks ◽  
Large Class ◽  
Lipschitz Function ◽  
Generation Gap

We consider a large class of random walks on the discrete circle Z/(n), defined in terms of a piecewise Lipschitz function, and motivated by the ‘generation gap' process of Diaconis. For such walks, we show that the time until convergence to stationarity is bounded independently of n. Our techniques involve Fourier analysis and a comparison of the random walks on Z/(n) with a random walk on the continuous circle S1.

scholarly journals Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

2020 ◽  
Vol 18 (1) ◽  
pp. 1-9
Author(s):  
Carlo Mariconda ◽  
Giulia Treu
Keyword(s):  
Convex Function ◽  
Calculus Of Variations ◽  
Open Subset ◽  
Lipschitz Function ◽  
Lavrentiev Phenomenon ◽  
Smooth Functions ◽  
Bounded Open Subset ◽  
Admissible Functions

Abstract We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

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 A CONTINUITY CHARACTERIZATION OF ASPLUND SPACES

2011 ◽  
Vol 83 (3) ◽  
pp. 450-455
Author(s):  
J. R. GILES
Keyword(s):  
Banach Space ◽  
Asplund Space ◽  
Weakly Compact ◽  
Asplund Spaces ◽  
Subdifferential Mapping

AbstractA Banach space is an Asplund space if every continuous gauge has a point where the subdifferential mapping is Hausdorff weak upper semi-continuous with weakly compact image. This contributes towards the solution of a problem posed by Godefroy, Montesinos and Zizler.

scholarly journals On rectifiable measures in Carnot groups: representation

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Gioacchino Antonelli ◽  
Andrea Merlo
Keyword(s):  
Large Class ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Lipschitz Functions ◽  
Carnot Groups ◽  
Geodesic Sphere ◽  
Area Formula ◽  
Definition Of ◽  
Lipschitz Graphs ◽  
Almost All

AbstractThis paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

scholarly journals Lipschitz functions with unexpectedly large sets of nondifferentiability points

2005 ◽  
Vol 2005 (4) ◽  
pp. 361-373 ◽  
Author(s):  
Marianna Csörnyei ◽  
David Preiss ◽  
Jaroslav Tišer
Keyword(s):  
Lipschitz Function ◽  
Measure Zero ◽  
Common Point ◽  
Lipschitz Functions ◽  
Large Sets ◽  
Set Of Points ◽  
Dense Set

It is known that everyGδsubsetEof the plane containing a dense set of lines, even if it has measure zero, has the property that every real-valued Lipschitz function onℝ2has a point of differentiability inE. Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct aGδsetE⊂ℝ2containing a dense set of lines for which there is a pair of real-valued Lipschitz functions onℝ2having no common point of differentiability inE, and there is a real-valued Lipschitz function onℝ2whose set of points of differentiability inEis uniformly purely unrectifiable.

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