关于$\mathbb R$ -线性映射的几何特征
On geometrical characterizations of $\mathbb R$-linear mappings
摘要：
我们考虑$\mathbb R$ -线性映射的几个特征。特别地，我们给出了值域为$\geq$ 2维的线性映射的一个特征，即该映射保线（和线到点的收缩）.这一刻划及其仿射形式推广了仿射几何基本定理。虽然$\mathbb R$ -线性映射作为可加函数的代数刻画依赖于集合论公理，但我们的结果在Zermelo公理系统（现代版本）中是可证明的，而不需要选择公理.
英文摘要：
We consider several characterizations of $\mathbb R$-linear mappings. In particular, we give a characterization of linear mappings whose range is $\geq$ 2 dimensional, in terms of preservation of lines (and contraction of lines to a point) by the mappings. This characterization and its affine version generalize the Fundamental Theorem of Affine Geometry. While the algebraic characterization of $\mathbb R$-linear mappings as additive functions depend on the axiom of set theory, our results are provable in (the modern version of) Zermelo's axiom system without Axiom of Choice.
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