Separable reduction theorems by the method of elementary submodels

We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to a special separable subspace, dependent only on the given set (function). We are interested in properties of sets "to be dense, nowhere dense, meager, residual or porous" and in properties of functions "to be continuous, semicontinuous or Fr\'echet differentiable". Our method of creating separable subspaces enables us to combine our results, so we easily get separable reductions of function properties such as "be continuous on a dense subset", "be Fr\'echet differentiable on a residual subset", etc. Finally, we show some applications of presented separable reduction theorems and demonstrate that some results of Zajicek, Lindenstrauss and Preiss hold in nonseparable setting as well.Comment: 27 page

Large separated sets of unit vectors in Banach spaces of continuous functions

The paper concerns the problem whether a nonseparable \C(K) space must contain a set of unit vectors whose cardinality equals to the density of \C(K) such that the distances between every two distinct vectors are always greater than one. We prove that this is the case if the density is at most continuum and we prove that for several classes of \C(K) spaces (of arbitrary density) it is even possible to find such a set which is $2$-equilateral; that is, the distance between every two distinct vectors is exactly 2.Comment: The second version does not contain new results, but it is reorganized in order to distinguish our main contributions from what was essentially know

Note on Bessaga-Klee classification

We collect several variants of the proof of the third case of the Bessaga-Klee relative classification of closed convex bodies in topological vector spaces. We were motivated by the fact that we have not found anywhere in the literature a complete correct proof. In particular, we point out an error in the proof given in the book of C.~Bessaga and A.~Pe\l czy\'nski (1975). We further provide a simplified version of T.~Dobrowolski's proof of the smooth classification of smooth convex bodies in Banach spaces which works simultaneously in the topological case.Comment: 14 pages; we made few corrections, added one reference and precised the abstrac

Isomorphisms between spaces of Lipschitz functions

We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that $\operatorname{Lip}_0(\mathbb{Z}^d)\simeq\operatorname{Lip}_0(\mathbb{R}^d)$, for all $d\in\mathbb{N}$. More generally, we e.g. show that $\operatorname{Lip}_0(\Gamma)\simeq \operatorname{Lip}_0(G)$, where $\Gamma$ is from a large class of finitely generated nilpotent groups and $G$ is its Mal'cev closure; or that $\operatorname{Lip}_0(\ell_p)\simeq\operatorname{Lip}_0(L_p)$, for all $1\leq p<\infty$. We leave a large area for further possible research.Comment: 28 pages, no figures. Accepted to Journal of Functional Analysi

Complexity of distances: Theory of generalized analytic equivalence relations

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov-Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that $E_1$ is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben-Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov-Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided.Comment: Based on the feedback we received, we decided to split the original version into two parts. The new version is now the first part of this spli

Projections in Lipschitz-free spaces induced by group actions

We show that given a compact group $G$ acting continuously on a metric space $M$ by bi-Lipschitz bijections with uniformly bounded norms, the Lipschitz-free space over the space of orbits $M/G$ (endowed with Hausdorff distance) is complemented in the Lipschitz-free space over $M$. We also investigate the more general case when $G$ is amenable, locally compact or SIN and its action has bounded orbits. Then we get that the space of Lipschitz functions $Lip_0(M/G)$ is complemented in $Lip_0(M)$. Moreover, if the Lipschitz-free space over $M$, $F(M)$, is complemented in its bidual, several sufficient conditions on when $F(M/G)$ is complemented in $F(M)$ are given. Some applications are discussed. The paper contains preliminaries on projections induced by actions of amenable groups on general Banach spaces