On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function

Luděk Zajíček

Displaying similar documents to “On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function”

A note on intermediate differentiability of Lipschitz functions

Luděk Zajíček (1999)

Commentationes Mathematicae Universitatis Carolinae

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Let $f$ be a Lipschitz function on a superreflexive Banach space $X$ . We prove that then the set of points of $X$ at which $f$ has no intermediate derivative is not only a first category set (which was proved by M. Fabian and D. Preiss for much more general spaces $X$ ), but it is even $σ$ -porous in a rather strong sense. In fact, we prove the result even for a stronger notion of uniform intermediate derivative which was defined by J.R. Giles and S. Sciffer.

Biseparating maps on generalized Lipschitz spaces

Denny H. Leung (2010)

Studia Mathematica

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Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F-valued functions on Y is said to be biseparating if f and g are disjoint if and only if Tf and Tg are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are...

A Lipschitz function which is $C^{\infty}$ on a.e. line need not be generically differentiable

Luděk Zajíček (2013)

Colloquium Mathematicae

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We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is $C^{\infty}$ smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially...

A new proof of Fréchet differentiability of Lipschitz functions

Joram Lindenstrauss, David Preiss (2000)

Journal of the European Mathematical Society

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We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on $ℓ_{2}$ (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the $w^{*}$ closure of the set of its points of Gâteaux differentiability is norm separable.

On continuous composition operators

Wilhelmina Smajdor (2010)

Annales Polonici Mathematici

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Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that $M_{ψ} : = s u p | | [r, s, t; ψ] | | : r < s < t, r, s, t \in I < \infty$ , where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ’ is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set (I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove...

Lipschitz and uniform embeddings into $ℓ_{\infty}$

N. J. Kalton (2011)

Fundamenta Mathematicae

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We show that there is no uniformly continuous selection of the quotient map $Q : ℓ_{\infty} \to ℓ_{\infty} / c ₀$ relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from $B_{X * *}$ onto $B_{X}$ .

Two applications of smoothness in C(K) spaces

Matías Raja (2014)

Studia Mathematica

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A simple observation about embeddings of smooth Banach spaces into C(K) spaces allows us to construct a parametrization of the separable Banach spaces using closed subsets of the interval [0,1]. The same idea is applied to the study of the isometric embedding of $ℓ_{p}$ spaces into certain C(K) spaces with the additional condition that the functions of the image must be Lipschitz with respect to a fixed finer metric on K. The feasibility of that kind of embeddings is related to Szlenk indices. ...

$P^{λ}$ -spaces and $L^{λ}$ -spaces

Nguyen To Nhu (1979)

Colloquium Mathematicae

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Double sine series with nonnegative coefficients and Lipschitz classes

Vanda Fülöp (2006)

Colloquium Mathematicae

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Denote by $f_{s s} (x, y)$ the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that $f_{s s}$ belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.

Singular points of order $k$ of Clarke regular and arbitrary functions

Luděk Zajíček (2012)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a separable Banach space and $f$ a locally Lipschitz real function on $X$ . For $k \in ℕ$ , let $Σ_{k} (f)$ be the set of points $x \in X$ , at which the Clarke subdifferential $\partial^{C} f (x)$ is at least $k$ -dimensional. It is well-known that if $f$ is convex or semiconvex (semiconcave), then $Σ_{k} (f)$ can be covered by countably many Lipschitz surfaces of codimension $k$ . We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe,...

On $r$ -reflexive Banach spaces

Iryna Banakh, Taras O. Banakh, Elena Riss (2009)

Commentationes Mathematicae Universitatis Carolinae

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A Banach space $X$ is called if for any cover $𝒰$ of $X$ by weakly open sets there is a finite subfamily $𝒱 \subset 𝒰$ covering some ball of radius 1 centered at a point $x$ with $∥ x ∥ \leq r$ . We prove that an infinite-dimensional separable Banach space $X$ is $\infty$ -reflexive ( $r$ -reflexive for some $r \in ℕ$ ) if and only if each $ε$ -net for $X$ has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of $X$ . We show that the quasireflexive James space $J$ is $r$ -reflexive for no $r \in ℕ$ . We do not know...

Lipschitz extensions of convex-valued maps

Alberto Bressan, Agostino Cortesi (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz $M$ , definita su un sottoinsieme di uno spazio di Hilbert $H$ a valori compatti e convessi in $\mathbb{R}^{n}$ , può essere estesa su tutto $H$ ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz $M$ .