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Carry out �standard� constructions in (linear) functional analysis: Abstract Hilbert space � one in each dimension, Concrete Hilbert space � Many, such as L2([0, 1])

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 $X$.

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincar\'e inequality. Such a functional is defined through relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean case. As an application we show that in a variational minimization problem related to the functional, boundary values can be presented as a penalty term.

We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $\sigma Z_\omega$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets $A+A+K$ and $A-A+K$ have non-empty interior in the completion $\bar X$ of $X$. This implies two results: (i) an analytic subgroup $A$ of a linear metric space $X$ is a $\sigma Z_\omega$-space if $A$ is not Polish and $A$ contains a Polish convex set $K$ with dense affine hull in $X$; (ii) a dense convex analytic subset $A$ of a linear metric space $X$ is a $\sigma Z_\omega$-space if $A$ contains no open Polish subspace and $A$ contains a Polish convex set $K$ with dense affine hull in $X$.

We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $\sigma Z_\omega$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets $A+A+K$ and $A-A+K$ have non-empty interior in the completion $\bar X$ of $X$. This implies two results: (i) an analytic subgroup $A$ of a linear metric space $X$ is a $\sigma Z_\omega$-space if $A$ is not Polish and $A$ contains a Polish convex set $K$ with dense affine hull in $X$; (ii) a dense convex analytic subset $A$ of a linear metric space $X$ is a $\sigma Z_\omega$-space if $A$ contains no open Polish subspace and $A$ contains a Polish convex set $K$ with dense affine hull in $X$.

We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $\sigma Z_\omega$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets $A+A+K$ and $A-A+K$ have non-empty interior in the completion $\bar X$ of $X$. This implies two results: (i) an analytic subgroup $A$ of a linear metric space $X$ is a $\sigma Z_\omega$-space if $A$ is not Polish and $A$ contains a Polish convex set $K$ with dense affine hull in $X$; (ii) a dense convex analytic subset $A$ of a linear metric space $X$ is a $\sigma Z_\omega$-space if $A$ contains no open Polish subspace and $A$ contains a Polish convex set $K$ with dense affine hull in $X$.

We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor controlled by \lambda(M). We prove that \lambda(M) is finite for several important classes of metric spaces. These include metric trees of arbitrary cardinality, groups of polynomial growth, Gromov-hyperbolic groups, certain classes of Riemannian manifolds of bounded geometry and finite direct sums of arbitrary combinations of these objects. On the other hand we construct an example of a two-dimensional Riemannian manifold M of bounded geometry for which \lambda(M)=\infty.

In this work we present a general theorem concerning chain rules for linear openness of set-valued mappings acting between metric spaces. As particular cases, we obtain classical and also some new results in this field of research, including the celebrated Lyusternik-Graves Theorem. The applications deal with the study of the well-posedness of the solution mappings associated to parametric variational systems. Sharp estimates for the involved regularity moduli are given.

In this paper, with a view to improve the g-monotonicity condition, we introduce the notion of g-comparability of a mapping defined on an ordered set and utilize the same to prove some existence and uniqueness results on coincidence points for linear contraction without g-monotonicity in ordered metric spaces. Our results extend some classical and well known results due to Ran and Reurings (Proc. Amer. Math. Soc. 132(2004), no.5, 1435-1443), Nieto and Rodriguez-Lopez (Acta Math. Sin. 23(2007), no.12, 2205-2212), Turinici (Libertas Math. 31(2011), 49-55), Turinici (Math. Student 81(2012), no.1-4, 219-229) and Doric et al. (RACSAM 108(2014), no.2, 503-510) and similar others.

In many real-world applications data come as discrete metric spaces sampled around 1-dimensional filamentary structures that can be seen as metric graphs. In this paper we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated, with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and we provide an efficient and easy to implement algorithm to compute such approximations in almost linear time. We illustrate the performances of our algorithm on a few synthetic and real data sets.

In many real-world applications data come as discrete metric spaces sampled around 1-dimensional filamentary structures that can be seen as metric graphs. In this paper we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated, with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and we provide an efficient and easy to implement algorithm to compute such approximations in almost linear time. We illustrate the performances of our algorithm on a few synthetic and real data sets.

The aim of this paper is to provide a comprehensive study of some linear nonlocal diffusion problems in metric measure spaces. These include, for example, open subsets in $\mathbb{R}^N$, graphs, manifolds, multi-structures or some fractal sets. For this, we study regularity, compactness, positiveness and the spectrum of the stationary nonlocal operator. Then we study the solutions of linear evolution nonlocal diffusion problems, with emphasis in similarities and differences with the standard heat equation in smooth domains. In particular prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.