论文标题
关于$ \ mathbb r $ - 线性映射的几何特征
On geometrical characterizations of $\mathbb R$-linear mappings
论文作者
论文摘要
我们考虑了$ \ Mathbb R $ - 线性映射的几种特征。特别是，我们给出了线性映射的特征，其范围为$ \ geq $ 2维度，就映射保存（以及线路的收缩到点的收缩）而言。这种特征及其仿射版本概括了仿射几何学的基本定理。尽管$ \ Mathbb r $ - 线性映射作为加法函数的代数表征取决于集合理论的公理，但我们的结果在没有选择的无公理的Zermelo的Axiom系统（现代版本）中得到证明。
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.
