等变Sard-Smale定理

The Equivariant SardSmales Theorem

在紧致李群Ｇ作用下的Ｂａｎａｃｈ空间Ｘ中，通过紧摄动方法和有限维逼近技巧，克服了现有结论中对空间的光滑性限制和对映射的全连续限制，Ｓａｒｄ－Ｓｍａｌｅ定理被推广为一致等变形式．设ｆ∈Ｃ１（Ｘ，Ｘ）是Ｇ－Ｆｒｅｄｈｏｌｍ映射，则对任意的ε＞０，都存在Ｇ等变全连续映射α∈Ｃ１（Ｘ，Ｘ）。

In the Banach space X acted by the compact Lie group G, the equivariant SardSmales theorem is proved. The difficulty is to overcome the lack of the smoothed norm of the Banach space. The methods used for this purpose are the compact perturbation and the finite dimensional approximation. The main result is as follows.Let f∈C1(X,X) be a GFredholm map. Then for any ε>0,there exists a compact map α∈C1(X,X), such that the zeroorbits of f+α are all regular and ‖α‖<ε.