赋范空间B(X,Y)与赋范空间K^(m×n)的等距性

Isometric property between normed space B(X,Y) and K^(m×n)

证明了存在X,Y,Km×n上的一组范数,使得数域K上的赋范线性空间B(X,Y)与赋范线性空间Km×n是等距同构的,n维内积空间X上的线性算子空间B(X,X)与(Kn×n,‖.‖2)是等距同构的,讨论了有限维空间上的线性算子的特征值与其对应矩阵的特征值的相互关系,有限维内积空间上的Hermite算子与Hermite矩阵间的相互关系.

It is proved that there are a set of norms in X,Y and K^m×n which make the normed linear space B（X,Y） in number field K and normed linear space K^m×n be isometric isomorphic, linear operator space B（X,Y） where X is a inner product space of n-dimension and （K^m×n, ｜｜ ·｜｜ 2） be isometric isomorphic. The mutual relation of the eigen- values of linear operator in finite-dimensional space and eigenvalues of its matrix, the mutual relation of Hermite op- erators in finite-dimensional inner product space and Hermite matrices are discussed.