The Serre closure of Banach spaces
Abstract
We determine the Serre closure of Banach spaces in the category of quasi-Banach spaces, that is, the smallest class of quasi-Banach spaces that contains all Banach spaces and is stable under subspaces, quotients, and extensions. Precisely, a quasi-Banach space X belongs to this closure if and only if there exist constants M and k∈ℕ such that (|x1+…+xn|)/(|x1|+…+|xn|)≤ Mlogk n for every n≥ 2 and every x1,…,xn∈ X; equivalently, X is a quotient of an Orlicz sequence space ℓ(I,fl), where I is an index set and fl(t) is equivalent to tlogl(1/t) near zero. The proof combines geometrical estimates for quasi-Banach spaces and quasilinear maps with the study of higher order derived spaces associated to complex interpolation between ℓ2/3 and ℓ2. We provide examples of spaces in the Serre closure of Banach spaces that are not iterated extensions of Banach spaces and discuss some related problems.
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BibTeX
@misc{sanchez2026serre,
title = {{The Serre closure of Banach spaces}},
author = {Félix Cabello Sánchez and Jesús M. F. Castillo and Raúl Martínez-Bohórquez and José Navarro Garmendia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35259},
eprint = {2610.35259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35259},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Serre closure of Banach spaces
AU - Félix Cabello Sánchez
AU - Jesús M. F. Castillo
AU - Raúl Martínez-Bohórquez
AU - José Navarro Garmendia
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35259
UR - https://arxiv.org/abs/2610.35259
AB - We determine the Serre closure of Banach spaces in the category of quasi-Banach spaces, that is, the smallest class of quasi-Banach spaces that contains all Banach spaces and is stable under subspaces, quotients, and extensions. Precisely, a quasi-Banach space X belongs to this closure if and only if there exist constants M and k∈ℕ such that (|x_1+…+x_n|)/(|x_1|+…+|x_n|)≤ Mlog^k n for every n≥ 2 and every x_1,…,x_n∈ X; equivalently, X is a quotient of an Orlicz sequence space ℓ(I,f_l), where I is an index set and f_l(t) is equivalent to tlog^l(1/t) near zero. The proof combines geometrical estimates for quasi-Banach spaces and quasilinear maps with the study of higher order derived spaces associated to complex interpolation between ℓ_2/3 and ℓ_2. We provide examples of spaces in the Serre closure of Banach spaces that are not iterated extensions of Banach spaces and discuss some related problems.
ER -