AI 中文总结
本文证明 Young 变差情形的 Levin-Milman 定理：闭线性子空间中元素在某个尺度下具有有限 Young 变差则子空间有限维，并由此得出无限维子空间中有限变差函数集的贫集性质及极大稠密可线性化结论。
AI 中文摘要
1940 年，Levin 和 Milman 证明了：$C[0,1]$ 的一个闭线性子空间，若其所有元素都具有有界 Jordan 变差，则必为有限维。我们证明了该定理在 Young 意义下的变差情形的类比，并且更一般地，对每个有限值、非降的规范函数 $\varphi:[0,\infty)\to[0,\infty)$（满足 $\varphi(0)=0<\varphi(t)$（对 $t>0$））均成立。若 $E$ 是 $C[0,1]$ 的闭线性子空间，且每个 $f\in E$ 在某个尺度 $\lambda_f>0$ 下满足 $\Var_\varphi(\lambda_f f)<\infty$，则 $E$ 必为有限维。无需连续性、凸性或倍增条件。证明结合了保持缩放类的下半连续正则化、Baire 均匀化、Helly 选择，以及针对非齐次规范函数的定量嵌套峰构造。因此，对每个无限维闭子空间 $F\subset C[0,1]$，集合 $F\cap\cV_\varphi([0,1])$ 是 $F$ 中的贫集 $F_\sigma$ 子集。对每个 Young 函数，缩放和原始有限变差族都是极大稠密可线性化的，但不是可空间化的。缩放族本身是 Hamel 维数为 $\mathfrak c$ 的稠密向量子空间，而若无局部倍增条件，原始族本身不必是线性的。
英文摘要
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 $φ:[0,\infty)\to[0,\infty)$ with $φ(0)=0<φ(t)$ for $t>0$. If $E$ is a closed linear subspace of $C[0,1]$ and every $f\in E$ satisfies $\Var_φ(λ_f f)<\infty$ 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 $F\subset C[0,1]$, the set $F\cap\cV_φ([0,1])$ 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 $\mathfrak c$, whereas without local doubling the raw family need not itself be linear.
Comments19 pages
