Norm Is Equivalent Operator Lie Algebra

The equivalence of norms is been widely studied recently. First, by the definition of the equivalence of the norms on a normed space, we give the definition of the equivalence of two β norms, and prove that any two β norms are equivalent on a finite dimensionalβ normed space. By giving a counter example, we prove the new defined equivalence is not simply generalized, the equivalence condition of two norms is not fit for the equivalence of two β norms. Then, as a application of the equivalence of two β norms on a finite dimensional normed space, we give a method to calculate the Banach-Mazur distance d(X,Rn), where X is a finite dimensional normed space, and also, we estimate the Banach-Mazur distance d(X,Y) of two different finite dimensional normed spaces X, Y.Lie algebras and Banach spaces have a closed connection. We can introduce a norm‖·‖on a algebra structure L, if the norm‖·‖satisfies‖xy‖≤‖x‖‖y‖and L is complete with the norm‖·‖, then L is a Banach algebra. The bounded linear operators B(X) of a Banach space X is a Banach algebra, for any operators T1,T2∈B(X), define a lie product [T1,T2]=T1T2-T2T1, then B(X) is a lie algebra. We prove that any symmetric compact linear operator generate a finite dimensional lie algebra with a projection operator on a Hilbert space. And on Hilbert space, we prove any rotation operator which rotation is not kπ, can not generate a finite dimensional lie algebra with a compact operator. Hence, we answer two questions raised by Sun Shanli and Cao Peng.This paper is divided into three chapters:The first chapter introduce the definition of two β norms on a β normed space, and proves the equivalence of any two β norms on a β normed space.The second chapter introduces the application of the equivalence of norms, the application is to calculate and estimate the Banach-Mazur distance.The third chapter discuss the operators which can generate finite dimensional operator lie algebra with a compact operator.