关于线性空间定义中各公理的独立问题

On the Independence of each Axiom in the Definition of Linear Space

本文证明了在有理数域上线性空间定义中公理5和公理6不独立,同时给出了公理5、公理6在复数域上独立和公理2、公理8在任意数域上独立的例子,在理论上有一定意义。

The article has proved the non-independence of Axioms 5 and 6 in the definition of linear space on the rational number field. At the same time, it has presented to us some examples to show the independence of Axioms 5 and 6 on the complex number field and the independence of Axioms 2 and 8 on the arbitrary number field. Its theorehical significance has been shown.