开映像定理的改良证明研究

Research on improved proof of open mapping theorem

为研究方程Tx=y解的稳定性,开映像定理将问题转化为研究映像T能否将开集映为开集.针对定理的证明,不再依据较抽象的对称凸集的性质,而是通过概念的等价转化,利用Banach空间的完备性,采用集合的平移及其运算性质,结合非疏集的定义以及算子的有界线性性质,并且使用逐次逼近的方法进行推理证明,从而得出满足T是开映像的充分条件,进而得到使得方程Tx=y的解稳定的条件.

In order to study the stability of the solution of the equation Tx=y,the open mapping theorem transforms the problem into the study of whether the open set can be mapped by the mapping T.For the proof of the theorem,this paper no longer based on the properties of the more abstract symmetric convex set,but used the equivalent transformation of the concept,utilized the completeness of Banach space,used the translation of the set and its operation properties,and combined the definition of non-sparse sets and the bounded linear properties of operators.Furthermore,the method of successive approximation was used to prove the reasoning.It was concluded that satisfying T was a sufficient condition for the image to be opened,and the condition for stabilizing the solution of the equation Tx=y was obtained.