On the metric compactification of infinite-dimensional $\ell_{p}$ spaces

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.