On the Metric Compactification of Infinite Dimensional \(\ell_p\) Spaces
Abstract
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.
Type
Publication
Canadian Mathematical Bulletin, 62(3), 491–507
Status
Peer-reviewed
Open access
Main results
We characterize all metric functionals on infinite-dimensional \(\ell_p\) spaces, \(1\leq p<\infty\).
1. Metric functionals on \(\ell_1\)
\[ h(x)=\sum_{i\in I}\sigma_i x_i+\sum_{i\notin I}\bigl(|x_i-a_i|-|a_i|\bigr), \]where \(I\subseteq\mathbb N\), \(\sigma_i\in\{-1,1\}\) for \(i\in I\), and \(a_i\in\mathbb R\) for \(i\notin I\).
The internal functionals correspond to \(I=\varnothing\) and \(a\in\ell_1\).
2. Metric functionals on \(\ell_p\) with \(1 < p <\infty\)
\[ h(x)=\left(\|x-a\|_p^p+c^p-\|a\|_p^p\right)^{1/p}-c, \]where \(a\in\ell_p\) and \(c\geq\|a\|_p\,\), or
\[ h(x)=-\langle u,x\rangle, \]where \(u\in\ell_q\,\), with \(\,\|u\|_q\leq1\,\) and \(\,pq=p+q\).
The internal functionals correspond to \(c=\|a\|_p \) in the first family.
Authors
Research Scientist
I’m a Researcher in Mathematics. I’ve carried out research at Aalto University, Inria Saclay, and VTT. I’ve made contributions to metric geometry, functional analysis and optimization. I’m passionate about quantum computing, risk analysis and optimization algorithms.