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

TL;DR

This paper generalizes the concept of metric compactification to infinite-dimensional Banach spaces and provides a complete description for $ ext{ell}_p$ spaces and Hilbert spaces, expanding understanding of their boundary structures.

Contribution

It introduces a generalized framework for metric compactification applicable to infinite-dimensional Banach spaces and characterizes it explicitly for $ ext{ell}_p$ and Hilbert spaces.

Findings

Complete description of metric compactification for $ ext{ell}_p$ spaces.

Full characterization of metric compactification for Hilbert spaces.

Extension of metric compactification theory to infinite-dimensional settings.

Abstract

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