关于向量到向量解析函数的一个注记

A NOTE ON ANALYTIC FUNCTIONS ON VECTORS TO VECTORS

本文证明了如下定理:设f(x),g(x)分别是Banach空间X的连通开子集D到Hilbert空间Y和Z中的解析函数,且f,g的值域R_f和R_g所生成的闭线性流形分别是Y和Z,若对任意x∈D,有‖f(x)=‖g(x)‖那末存在唯一的有界可逆线性算子U:Y→Z,U保持内积,并且对任意x∈D,有Uf(x)=g(x).

In this paper, we prove the followihg theorem: Let, f(x)g(x) be analytic functions defined on a connected open subset D ofa Complex Banach space X to Hilbert Spaces Y and Z respecti-vely, Y and Z are the minimal closed linear varieties generated bythe ranges R_f and R_g of f, g, respectively. If for any x∈D, ||f(x)||=||g(x)||,then, there exists a unique bounded invertible linear operatorU:Y→Z, U preserves the inner Product. and for every x∈D, wehave Uf(x) = g(x)