A Levin-Milman theorem for Young variation
Abstract
In 1940, Levin\footnotemark[1] and Milman proved that a closed linear subspace of C[0,1] whose elements all have bounded Jordan variation must be finite-dimensional. We prove its analogue for variation in the sense of Young and, more generally, for every finite-valued nondecreasing gauge φ:[0,∞)→[0,∞) with φ(0)=0<φ(t) for t>0. If E is a closed linear subspace of C[0,1] and every f∈ E satisfies Varφ(λf f)<∞ at some scale λf>0, then E is finite-dimensional. No continuity, convexity, or doubling condition is needed. The proof combines a lower-semicontinuous regularization that preserves the scaled class, Baire uniformization, Helly selection, and a quantitative nested-peaks construction for nonhomogeneous gauges. Consequently, for every infinite-dimensional closed subspace F⊂ C[0,1], the set F∩cVφ([0,1]) is a meagre Fσ subset of F. For every Young function, both the scaled and raw finite-variation families are maximal dense-lineable but not spaceable. The scaled family is itself a dense vector subspace of Hamel dimension c, whereas without local doubling the raw family need not itself be linear.
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BibTeX
@misc{albuquerque2026levinmilman,
title = {{A Levin-Milman theorem for Young variation}},
author = {N. G. Albuquerque and D. Bugajewska and E. Demétrio Jr. and J. B. Seoane-Sepúlveda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07132},
eprint = {2609.07132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07132},
}
RIS
TY - UNPB
TI - A Levin-Milman theorem for Young variation
AU - N. G. Albuquerque
AU - D. Bugajewska
AU - E. Demétrio Jr.
AU - J. B. Seoane-Sepúlveda
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07132
UR - https://arxiv.org/abs/2609.07132
AB - In 1940, Levin\footnotemark[1] and Milman proved that a closed linear subspace of C[0,1] whose elements all have bounded Jordan variation must be finite-dimensional. We prove its analogue for variation in the sense of Young and, more generally, for every finite-valued nondecreasing gauge φ:[0,∞)→[0,∞) with φ(0)=0<φ(t) for t>0. If E is a closed linear subspace of C[0,1] and every f∈ E satisfies Var_φ(λ_f f)<∞ at some scale λ_f>0, then E is finite-dimensional. No continuity, convexity, or doubling condition is needed. The proof combines a lower-semicontinuous regularization that preserves the scaled class, Baire uniformization, Helly selection, and a quantitative nested-peaks construction for nonhomogeneous gauges. Consequently, for every infinite-dimensional closed subspace F⊂ C[0,1], the set F∩cV_φ([0,1]) is a meagre F_σ subset of F. For every Young function, both the scaled and raw finite-variation families are maximal dense-lineable but not spaceable. The scaled family is itself a dense vector subspace of Hamel dimension c, whereas without local doubling the raw family need not itself be linear.
ER -