中文摘要：

在向量空间上由半范数定义了一个拓扑，在有限维向量空间中，把范数的一些重要结论扩展到半范数，研究了半范数在它导出的拓扑空间上等价性的判定，以及连续性等重要结论，并给出了简化证明．

英文摘要：

In this paper, the author gives a topology defined by semi -norms on a vector space at first. After that, the author extends an important conclusion of norms on a finite - dimensional vector space to the semi - norms in the same case. The paper studies some properties of semi - norm space such as equivalence and continuity on the topology that is defined by semi - norm itself and makes simplified proof.