A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Frechet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called $Γ$-null sets), whose definition involves both the notions of category and measure, and showing that the required differentiability holds almost everywhere with respect to it. We even obtain existence of Fréchet derivatives of Lipschitz functions between certain infinite-dimensional Banach spaces;no such results have been known previously. Our main result states that a Lipschitz map between separable Banach spaces is Fréchet differentiable $Γ$-almost everywhere provided that it is regularly Gateaux differentiable $Γ$-almost everywhere and the Gateaux derivatives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of X to spaces with the Radon-Nikodym property are Gateaux differentiable $Γ$-almost everywhere. Moreover, Gateaux differentiability implies regular Gateaux differentiability with exception of another kind of negligible sets, so-called $σ$-porous sets. The answer to the question is therefore positive in every space in which every $σ$-porous set is $Γ$-null. We show that this holds for $C(K)$ with $K$ countable compact, the Tsirelson space and for all subspaces of $c_0$, but that it fails for Hilbert spaces.
Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X. Let Pc from X to C denote the (standard) metric projection operator. In this paper, we define the Gateaux directional differentiability of Pc. We investigate some properties of the Gateaux directional differentiability of Pc. In particular, if C is a closed ball or a closed and convex cone (including proper closed subspaces), then, we give the exact representations of the directional derivatives of Pc. Finally, we define the concept of p-q uniformly convex and uniformly smooth Ban…
Let $X_1, \dots, X_n$ be Banach spaces and $f$ a real function on $X=X_1 \times\dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which $f$ is partially Fr\' echet differentiable but is not Fr\' echet differentiable. Our results imply that if $X_1, \dots, X_{n-1}$ are Asplund spaces and $f$ is continuous (resp. Lipschitz) on $X$, then $A_f$ is a first category set (resp. a $σ$-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f:X \to Y$ is a Lipschitz mapping, then the set of all points $x \in X$ at which $f$ is G\^ ateaux differentiable, is Fr\…
In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact, limited and completely continuous linear operators. Moreover, our results yield a characterization of Gelfand-Phillips spaces and recover some known result of Schur spaces and reflexive spaces concerning the differentiability of real-valued Lipschitz…
We show that Sobolev maps with values in a dual Banach space can be characterized in terms of weak derivatives in a weak* sense. Since every metric space embeds isometrically into a dual Banach space, this implies a characterization of metric space valued Sobolev maps in terms of such derivatives. Furthermore, we investigate for which target spaces Sobolev maps are weak* differentiable almost everywhere.
We establish a necessary and sufficient condition for the differentiability of the distance function generated by a nonempty closed set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform differentiability constraints on the norm of the space as in Giles [16]. Hence, our result advances that of Giles [16]. We prove that the proximinal condition of Giles [16] is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A n…
A Geometry Characteristic for Banach Space with $c^1$-Norm
设$E$是一个巴拿赫空间，在$ E \backslash \{0\}$和$S(E)=\{e\in E: \|e\|=1\}.$中具有$c^1$ -范数$\|\cdot\|$本文利用$S(E).$的一个几何构造给出了$E$的一个几何特征，即下列定理成立：$E$的范数是$ E \backslash \{0\}$中的$c^1$的范数当且仅当$S(E)$是具有${\rm codim}S(E)=1.$的$E,$的$c^1$ -子流形该定理非常清楚，然而，它的证明是非平凡的，表明了$ E \backslash \{0\}$中范数$\|\cdot\|$的连续可微性与$S(E).$的微分结构之间的内在联系
Let $\mathcal{L}(X;Y)$ be the space of bounded linear operators from a Banach space $X$ to a Banach space $Y$. Given an operator-valued function $u:\mathbb{R}_{\geq 0}\rightarrow \mathcal{L}(X;Y)$, suppose that every orbit $t\mapsto u(t)x$ has a regularity property (e.g. continuity, differentiability, etc.) on some interval $(t_x,\infty)$ in general depending on $x\in X$. In this paper we develop an abstract set-up based on Baire-type arguments which allows, under certain conditions, removing the dependency on $x$ systematically. Afterwards, we apply this theoretical framework to several diffe…
Let $S$ be a closed subset of a Banach space $X$. Assuming that $S$ is epi-Lipschitzian at $\bar{x}$ in the boundary $ \bd S$ of $S$, we show that $S$ is strictly Hadamard differentiable at $\bar{x}$ IFF the Clarke tangent cone $T(S, \bar{x})$ to $S$ at $\bar{x}$ contains a closed hyperplane IFF the Clarke tangent cone $T(\bd S, \bar{x})$ to $\bd S$ at $\bar{x}$ is a closed hyperplane. Moreover when $X$ is of finite dimension, $Y$ is a Banach space and $g: X \mapsto Y$ is a locally Lipschitz mapping around $\bar{x}$, we show that $g$ is strictly Hadamard differentiable at $\bar{x}$ IFF $T(\mat…
A survey on the Asplund property for spaces $C(X)$ of continuous functions
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Faculty of Mathematics and Informatics, A. Mickiewicz University（亚当·密茨凯维奇大学数学与信息学院）；Department of Mathematics, Ben-Gurion University of the Negev（内盖夫本-古里安大学数学系）
