The notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel; and 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 infinitedimensional ℓp spaces for all 1 ≤ p < ∞. We also give a full characterization of the metric compactification of infinite-dimensional Hilbert spaces.2010 Mathematics Subject Classification. Primary 54D35, Secondary 53C23, 46B45.

We give a complete description of the horofunction boundary of finite-dimensional ℓ p spaces, for 1 ≤ p ≤ ∞. We also study the variation norm on R N , 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 R N + of R N .

We present a complete characterization of the metric compactification of L p spaces for 1 ≤ p < ∞. 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 < ∞, and Alspach's example of an isometry on a weakly compact convex subset of L 1 with no fixed points.

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 $$\ell ^{1}$$ ℓ 1 . We also point out some inaccurate assertions appearing in the literature on this topic.