THE EXTENSION AND APPLICATION OF THE EQUIVALENCE OF NORMS ON A FINITE DIMENSIONAL SPACE

Abstract

We introduce the concept of the equivalence of \beta-norms on a linear space which
can be given different \beta-norms completely. Then for every n-dimensional \beta-normed space X,
by using the norm of X: ||x|| = | ( \sum_{i=1}^{n} |\xi_i|^2 )^{\frac{1}{2}}, we get a new \beta-normed space
(X, ||x||^{\beta}), and get a conclusion that any \beta-norm on a nite dimensional \beta-normed space is
equivalent to ||x||^{\beta}. Further more, we prove that all of the \beta-norms on a nite dimensional
linear space are equivalent. At last, we give an application of norm equivalence: Suppose
X, Y are two n-dimensional real spaces, then the Banach-Mazur distance d(X, R^n) = c_1^{-1} c_2,
where c_1, c_2 are two constants concerned with the norm of X. We also give an estimation of d(X, Y ).