Eva Pernecká

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

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...

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...

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}}}^{...

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...

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}...

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.

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...

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,...

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...

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...

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...

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.

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...

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...

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...

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...

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.

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...

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.