arXiv subjects

Antonin Prochazka

Publications and source records attributed to Antonin Prochazka.

5 recordsLinked to original sources

arXiv2022

Injectivity of Lipschitz operators

Any Lipschitz map $f\colon M \to N$ between metric spaces can be "linearised" in such a way that it becomes a bounded linear operator $\widehat{f}\colon \mathcal F(M) \to \mathcal F(N)$ between the Lipschitz-free spaces over $M$ and $N$. The purpose of this note is to explore the connections between the injectivity of $f$ and the injectivity of $\widehat{f}$. While it is obvious that if $\widehat{f}$ is injective then so is $f$, the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces $M$, any injective Lipschitz map $f\colon M \to N$ (for any $N$) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on $f$ which ensure that $\widehat{f}$ is injective.

math.FA↗

arXiv2019

Pełczyński space is isomorphic to the Lipschitz free space over a compact set

We prove the result stated in the title. This provides a first example of an infinite-dimensional Banach space whose Lipschitz free space is isomorphic to the free space of a compact set.

math.FA↗

arXiv2016

Linear properties of Banach spaces and low distortion embeddings of metric graphs

We characterize non-reflexive Banach spaces by a low-distortion (resp. isometric) embeddability of a certain metric graph up to a renorming. Also we study non-linear sufficient conditions for $\ell_1^n$ being $(1+\varepsilon)$-isomorphic to a subspace of a Banach space $X$.

math.FA↗

arXiv2009

$C^k$-smooth approximations of LUR norms

Let $X$ be a WCG Banach space admitting a $C^k$-Fr\' echet smooth norm. Then $X$ admits an equivalent norm which is simultaneously $C^1$-Fr\' echet smooth, LUR, and a uniform limit of $C^k$-Fr\' echet smooth norms. If $X=C([0,α])$, where $α$ is an ordinal, then the same conclusion holds true with $k=\infty$.

math.FA↗

arXiv2009

Weak$^*$ dentability index of spaces $C([0,α])$

We compute the weak$^*$-dentability index of the spaces $C(K)$ where $K$ is a countable compact space. Namely ${Dz}(C([0,ω^{ω^α}])) = ω^{1+α+1}$, whenever $0\leα<ω_1$. More generally, ${Dz}(C(K))=ω^{1+α+1}$ if $K$ is a scattered compact whose height $η(K)$ satisfies $ω^α<η(K)\leq ω^{α+1}$ with an $α$ countable.

math.FA↗