Related papers: Weakly almost square Lipschitz-free spaces

We prove in particular that the Lipschitz-free space over a finitely-dimensional normed space is complemented in its bidual. For Euclidean spaces the norm of the respective projection is $1$. As a tool to obtain the main result we establish…

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism…

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…

We study transfinite analogues of the symmetric strong diameter two property. We investigate the stability of these properties under $c_0$, $\ell_\infty$ sums and under projective tensor products. Moreover, we characterize Banach spaces of…

In this paper we show that if $(y_n)$ is a seminormalized sequence in a Banach space which does not have any weakly convergent subsequence, then it contains a wide-$(s)$ subsequence $(x_n)$ which admits an equivalent convex basic sequence.…

We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain high regularity properties - having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a…

Let $k$ be a complete valuation field. We formulate a free Banach $k$-vector space as a Banach $k$-vector space with an orthonormal Schauder basis, and an almost free Banach $k$-vector space as a non-Archimedean analogue of an almost free…

We study quantitative versions of the Schur property and weak sequential completeness, proceeding thus with investigations started by G. Godefroy, N. Kalton and D. Li and continued by H. Pfitzner and the authors. We show that the Schur…

We prove that a closed convex subset of a Banach space is (super-)weakly compact if and only if it is (super)-ergodic. As a consequence we deduce that super weakly compact sets are characterized by the fixed point property for continuous…

We study the geometry of a weak Riemannian metric on the infinite dimensional manifold of compact spacelike Cauchy hypersurfaces in a globally hyperbolic spacetime. We show that the geodesic distance (i.e. the infimum of lengths of paths…

We continue the study initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006), no. 1, 65-86] of properties related to greedy bases in the case when the constants involved are sharp,…

Let $X\left(\mathbb{R}^{n}\right)$ be a ball quasi-Banach function space on $\mathbb{R}^{n}$, $WX\left(\mathbb{R}^{n}\right)$ be the weak ball quasi-Banach function space on $\mathbb{R}^{n}$, $H_{X}\left(\mathbb{R}^{n}\right)$ be the Hardy…

We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm…

We study the density of the set $\operatorname{SNA}(M,Y)$ of those Lipschitz maps from a (complete pointed) metric space $M$ to a Banach space $Y$ which strongly attain their norm (i.e.\ the supremum defining the Lipschitz norm is actually…

Let $X$ be a Banach space and let $C$ be a closed convex bounded subset of $X$. It is proved that $C$ is weakly compact if, and only if, $C$ has the {it generic} fixed point property ($\mathcal{G}$-FPP) for the class of $L$-bi-Lipschitz…

We investigate geometric properties of a metric measure space where every function in the Newton--Sobolev space $N^{1,\infty}(Z)$ has a Lipschitz representative. We prove that when the metric space is locally complete and the reference…

We present an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The…

There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the…

Let $\mathrm{Lip}_0(X)$ be the space of all Lipschitz scalar-valued functions on a pointed metric space $X$. We characterize the approximation property for $\mathrm{Lip}_0(X)$ with the bounded weak* topology using as tools the tensor…

In this paper we give a thorough study of Lipschitz spaces. We obtain the following new results: (1) Sharp Jawerth-Franke-type embeddings between the Besov and Lipschitz spaces extending the classical results for Besov and Sobolev spaces;…