Recognition and reconstruction of sets in \(\ell^2\) via their projections
type: Item
instance of: scholarly article
title: Recognition and reconstruction of sets in \(\ell^2\) via their projections
author: Stoyu Barov
published in: Topology Proceedings
publication date: 2024-05-29T00:00:00Z
review text: Let \(k\in \mathbb{N}\) and let \(\mathcal{P}\) be a subset of all \(k\)-dimensional linear subspaces \(\mathcal{G}_k\) of \(\ell^{2}\) with the natural topology. The subsets \(B\) and \(C\) of \(\ell^{2}\) are called \(\mathcal{P}\)-imitations of each other if \(B+P=C+P\) for every \(P\in \mathcal{P} \). The author points out that, for \(\mathcal{P}\) a somewhere dense \(\mathcal{G}_\delta\)-set in \(\mathcal{G}_k\), there are certain non-trivial sets in \(\ell^{2}\) such that each of them has only one \(\mathcal{P}\)-imitation, namely, itself and further every such set can be reconstructed as the intersection of the preimages of its projections under \(\mathcal{P}\). In addition, the author discusses some important properties of \(\sigma\)-compact sets that are of independent interest.
reviewed by: Uğur Şengül
Mathematics Subject Classification ID: 51A05
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 7856050
zbMATH Keywords: separable Hilbert space
zbMATH Keywords: orthogonal projection
zbMATH Keywords: set with empty geometric interior
zbMATH Keywords: reconstruction of a set
zbMATH Keywords: dense \(G_\delta\)-set
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7856050