On the metric compactification of infinite-dimensional ${\mathrm{\ell }}_p$ spaces

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 infinite-dimensional ${\mathrm{\ell }}_p$ spaces for all $1\le p<\mathrm{\infty }$. We also give a full characterization of the metric compactification of infinite-dimensional Hilbert spaces.