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Nonisomorphic separable Banach spaces at Lipschitz distance one
Abstract
We construct nonisomorphic separable real Banach spaces whose multiplicative Lipschitz distance is one. Building on OpenAI's graph-space construction, we arrange two sequences of graph spaces into countably many blocks and form two spaces that differ by a single additional block. This block distinguishes their linear structures: one space contains a complemented copy of ℓ1(c0(ℓ2)), whereas the other does not. By mapping the additional block to increasingly late positions and rearranging the remaining blocks, we obtain homeomorphisms whose Lipschitz constants and those of their inverses tend to one, while both spaces and their norms remain unchanged.
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BibTeX
@misc{tanaka2026nonisomorphic,
title = {{Nonisomorphic separable Banach spaces at Lipschitz distance one}},
author = {Ryotaro Tanaka},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06493},
eprint = {2611.06493},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06493},
}
RIS
TY - UNPB
TI - Nonisomorphic separable Banach spaces at Lipschitz distance one
AU - Ryotaro Tanaka
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06493
UR - https://arxiv.org/abs/2611.06493
AB - We construct nonisomorphic separable real Banach spaces whose multiplicative Lipschitz distance is one. Building on OpenAI's graph-space construction, we arrange two sequences of graph spaces into countably many blocks and form two spaces that differ by a single additional block. This block distinguishes their linear structures: one space contains a complemented copy of ℓ_1(c_0(ℓ_2)), whereas the other does not. By mapping the additional block to increasingly late positions and rearranging the remaining blocks, we obtain homeomorphisms whose Lipschitz constants and those of their inverses tend to one, while both spaces and their norms remain unchanged.
ER -