线性空间中函数的单位分解性

On the Unit Decomposition of Functions in the Linear Space

利用微分流形研究了实数域R上的n维线性空间X的连续函数,指出在线性空间X中存在满足单位分解的一簇光滑函数,并得到了线性空间X中函数的一个单位分解定理;另外,在二维线性空间X中应用单位分解定理,指出了存在定义在任意开凸体A上,满足单位分解性的3个连续函数,这样将线性空间与微分流形相结合,为研究线性空间的函数分解问题及相关问题提出了一种新方法。

Using the knowledge of differential manifold, the linear space with the dimension of n on the real numberfield was researched. In linear space , the existence about a cluster of smooth functions satisfyingthe unit decomposition were pointed out, and a unit decomposition theorem was established. In addition, applying theunit decomposition theorem in the linear space with the dimension of 2, the existence was shown that the threecontinuous functions satisfying the unit decomposition were defined in arbitrary open convex body . The linear spacewas combined with differential manifold, and a new method was showed to research decomposition and relatedproblem of functions in the general linear space.