Recognition and reconstruction of sets in $\ell^2$ via their projections
Keywords:
separable Hilbert space, orthogonal projection, set with empty geometric interior, reconstruction of a set, dense $G_\delta$-setAbstract
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 $P \in \mathcal{P}$. In the case when $\mathcal{P}$ is somewhere dense $G_\delta$-set in $\mathcal{G}_k$, we show that there are certain non-trivial sets in $\ell^2$ such that each of them has only one $\mathcal{P}$-imitation, namely, itself. Consequently, every such set can be reconstructed as the intersection of the preimages of its projections under $\mathcal{P}$. In addition, we discuss some important properties of $\sigma$-compact sets that are of independent interest.
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