We introduce the notion of a firm non-expansive mapping in weak metric spaces, extending previous work for Banach spaces and certain geodesic spaces. We prove that, for firm non-expansive mappings, the minimal displacement, the linear rate of escape, and the asymptotic step size are all equal. This generalises a theorem by Reich and Shafrir.

This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of $\ell_{1}$. We also point out some inaccurate assertions appearing in the literature on this topic.

We present a complete characterization of the metric compactification of $L_p$ spaces for $1 \leq p < \infty$. Each element of the metric compactification of $L_p$ is represented by a random measure on a certain Polish space. By way of illustration, we revisit the $L_p$-mean ergodic theorem for $1 < p < \infty$, and Alspach’s example of an isometry on a weakly compact convex subset of $L_1$ with no fixed points.

The notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel. It has been mainly studied on proper geodesic metric spaces. We present here a generalization of the metric compactification that can be applied to infinite-dimensional Banach spaces. Thereafter we give a complete description of the metric compactification of infinite-dimensional $\ell_p$ spaces for all $1 \leq p < \infty$. We also give a full characterization of the metric compactification of infinite-dimensional Hilbert spaces.

We give a complete description of the horofunction boundary of finite-dimensional $\ell_{p}$ spaces for $1 \leq p \leq \infty$. We also study the variation norm on $\mathbf{R}^\mathcal{N}$, $\mathcal{N}=\{1,…,N\}$, and the corresponding horofunction boundary. As a consequence, we describe the horofunctions for Hilbert’s projective metric on the interior of the standard cone $\mathbf{R}^\mathcal{N}_{+}$ of $\mathbf{R}^\mathcal{N}$.