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 $\varphi:[0,\infty)\to[0,\infty)$ with $\varphi(0)=0<\varphi(t)$ for $t>0$. If $E$ is a closed linear subspace of $C[0,1]$ and every $f\in E$ satisfies $\Var_\varphi(\lambda_f f)<\infty$ at some scale $\lambda_f>0$, 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 $F\subset C[0,1]$, the set $F\cap\cV_\varphi([0,1])$ is a meagre $F_\sigma$ 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 $\mathfrak c$, whereas without local doubling the raw family need not itself be linear.