Eva Pernecká

Eva Pernecká
Czech Technical University in Prague | ČVUT · Faculty of Information Technology (FIT)

About

20
Publications
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187
Citations
Introduction

Publications

Publications (20)
Article
Full-text available
We analyze the relationship between Borel measures and continuous linear functionals on the space $\textrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak$^\ast $ continuous functionals, that is, members of the Lipschitz-fre...
Article
Full-text available
Let Lip0(M) be the space of Lipschitz functions on a complete metric space M that vanish at a base point. We prove that every normal functional in Lip0(M)* is weak* continuous; that is, in order to verify weak* continuity it suffices to do so for bounded monotone nets of Lipschitz functions. This solves a problem posed by N. Weaver. As an auxiliary...
Article
Given a topological group G that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that G has invariant linear span if all linear spans of G under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set A let \({{\mathbb {Z}}}^{...
Preprint
Full-text available
We analyze the relationship between Borel measures and continuous linear functionals on the space Lip0(M) of Lipschitz functions on a complete metric space M. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak* continuous functionals, i.e. members of the Lipschitz-free space F(M), measures on...
Preprint
Given a topological group $G$ that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that $G$ has invariant linear span if all linear spans of $G$ under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set $A$ let $\mathbb{Z}...
Preprint
Full-text available
Let $\operatorname{Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space $M$ that vanish at a base point. We show that every normal functional in $\operatorname{Lip}_0(M)^\ast$ is weak$^*$ continuous, answering a question by N. Weaver.
Article
Full-text available
We show that the class of Lipschitz-free spaces over closed subsets of any complete metric space M is closed under arbitrary intersections, improving upon the previously known finite-diameter case. This allows us to formulate a general and natural definition of supports for elements in a Lipschitz-free space F(M). We then use this concept to study...
Article
Full-text available
For a complete metric space M, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space F(M) are precisely the elementary molecules (δ(p)−δ(q))/d(p, q) defined by pairs of points p, q in M such that the triangle inequality d(p, q) < d(p,r) + d(q, r) is strict for any r ∈ M different from p and q. To this end,...
Preprint
We show that the class of Lipschitz-free spaces over closed subsets of any complete metric space $M$ is closed under arbitrary intersections, improving upon the previously known finite-diameter case. This allows us to formulate a general and natural definition of supports for elements in a Lipschitz-free space $\mathcal F(M)$. We then use this conc...
Preprint
Full-text available
We show that the class of Lipschitz-free spaces over closed subsets of any complete metric space M is closed under arbitrary intersections, improving upon the previously known finite-diameter case. This allows us to formulate a general and natural definition of supports for elements in a Lipschitz-free space F (M). We then use this concept to study...
Preprint
Full-text available
For a complete metric space $M$, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space $\mathcal{F}(M)$ are precisely the elementary molecules $(\delta(p)-\delta(q))/d(p,q)$ defined by pairs of points $p,q$ in $M$ such that the triangle inequality $d(p,q)<d(p,r)+d(q,r)$ is strict for any $r\in M$ different...
Article
Full-text available
Let $M$ be a compact subset of a superreflexive Banach space. We prove a certain `weak$^\ast$-version' of Pe\l czy\'nski's property (V) for the Banach space of Lipschitz functions on $M$. As a consequence, we show that its predual, the Lipschitz-free space $\mathcal{F}(M)$, is weakly sequentially complete.
Article
We prove that for any separable Banach space X, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to X. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails th...
Article
In this note we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property. This answers a question by G. Godefroy. We also prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise no...
Article
In 1970 Haskell Rosenthal proved that if $X$ is a Banach space, $\Gamma$ is an infinite index set, and $T:\ell_\infty(\Gamma)\to X$ is a bounded linear operator such that $\inf_{\gamma\in\Gamma}\|T(e_\gamma)\|>0$ then $T$ acts as an isomorphism on $\ell_\infty(\Gamma')$, for some $\Gamma'\subset\Gamma$ of the same cardinality as $\Gamma$. Our main...
Article
We prove that for certain subsets $M \subseteq \mathbb{R}^N$, $N \geqslant 1$, the Lipschitz-free space $\mathcal{F}(M)$ has the metric approximation property (MAP), with respect to any norm on $\mathbb{R}^N$. In particular, $\mathcal{F}(M)$ has the MAP whenever $M$ is a finite-dimensional compact convex set. This should be compared with a recent r...
Article
The main result implies that the Lipschitz-free spaces F(ℓ1)F(ℓ1) and F(Rn)F(Rn) have a Schauder basis. This improves (in a special case) on the previous work of Godefroy and Kalton who showed that F(X)F(X) has a bounded approximation property if and only if the Banach space X does.
Article
We study compactness properties of Hardy operators involving suprema on weighted Banach function spaces. We first characterize the compactness of abstract operators assumed to have their range in the class of non-negative monotone functions. We then define a category of pairs of weighted Banach function spaces for which a suitable Muckenhoupt-type...
Article
We prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. We also show that the Lipschitz-free spaces over $\ell_1^N$ or $\ell_1$ have monotone finite-dimensional Schauder decompositions.

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Project (1)
Project
My part in the project is to investigate Banach space structure of Lipschitz-free spaces, with particular focus on the so-called property (X) connected with the unique predual problem. I also study some Ulam-type approximation problems in operator algebras.