中文摘要：

推广了欧氏空间中的相应概念,给出了Banach空间中实函数芽的K-bi-Lipschitz等价,K-M-bi-Lipschitz等价与相同的切触概念,得到了判别可分的Banach空间上的两个实函数芽为C-bi-Lipschitz等价的一种行之有效的方法.同时指出K-bi-Lipschitz等价与K-M-bi-Lipschitz等价之间的关系.并给出Banach空间上的多项式函数芽在K-bi-Lipschitz等价关系下进行分类的基础.

英文摘要：

The notions of K-bi-Lipschitz equivalence,the same contact and K-M-bi-Lipschitz equivalence between real function germs in Euclidean spaces are generalized to Banach spaces.The criterion method of C-bi-Lipschitz equivalence between real function germs in separable Banach spaces is given.The relationship of K-bi-Lipschitz equivalence and K-M-bi-Lipschitz equivalence is investigated.The main results provide the basis of the classification of polynomial function germs on Banach space under K-bi-Lipschitz equivalence relation.