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A question of Ulam and a theorem of de Rham
Abstract
We generalize de Rham's decomposition theorem to proper metric spaces: every proper metric space has a unique decomposition as the product of a Euclidean space and a finite or countable pointed ℓ2-product of nontrivial irreducible non-Euclidean factors. More generally, every complete metric space splits canonically into a Hilbert factor, a pointed ℓ2-product of irreducible non-Euclidean factors, and a factor with no nontrivial irreducible factors. These results partially answer a question of Ulam.
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BibTeX
@misc{lytchak2026question,
title = {{A question of Ulam and a theorem of de Rham}},
author = {Alexander Lytchak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00200},
eprint = {2610.00200},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00200},
}
RIS
TY - UNPB
TI - A question of Ulam and a theorem of de Rham
AU - Alexander Lytchak
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00200
UR - https://arxiv.org/abs/2610.00200
AB - We generalize de Rham's decomposition theorem to proper metric spaces: every proper metric space has a unique decomposition as the product of a Euclidean space and a finite or countable pointed ℓ^2-product of nontrivial irreducible non-Euclidean factors. More generally, every complete metric space splits canonically into a Hilbert factor, a pointed ℓ^2-product of irreducible non-Euclidean factors, and a factor with no nontrivial irreducible factors. These results partially answer a question of Ulam.
ER -