F-空间中的一个基本定理及其在数值分析中的应用（英文）

A FUNDAMENTAL THEOREM ON F-SPACES AND ITS APPLICATION IN NUMERICAL ANALYSIS

本文针对F-空间中闭算子方程的一般逼近格式,研究其相容性、收敛性和稳定性之间的关系.所得的主要结果是:这种一般逼近格式在相容性条件下,其收敛性与稳定性是等价的.此定理可以看作是对Lax等价原理的推广,是求解第一类闭算子方程的一般逼近格式的基本定理.为得到这一主要结果,本文还给出了F-空间中的一条基本定理,众所周知的一致有界原理,闭图像定理和开映像定理是其简单推论.

The main results of this paper are Theorems 1.1 and 1.2 in Section 1. Theorem 1.1 may be regarded as the basic tool in the theory of F-spaces, for it implies fundamental principles such as the uniform boundedness theorem, the open mapping theorem, and the closed graph theorem. Theorem 1.2 can be viewed as an application of Theorem 1.1 in numerical analysis, which shows that an abstract approximation scheme （consisting of a sequence of closed operators with closed ranges in F-space setting） is convergent if and only if it is stable.