Norms induced by symmetric guage functions

Abstract

Let F n be the set of all n×1 column vectors over F, where F=R, A norm ∥ċ∥ on F n is permutationally invariant if and it is an absolute norm if A permutationally invariant absolute norm on F n is called a symmetric gauge function. Given a norm ∥ċ∥ on F n and a nonsingular matrix HεF n×n , one can define a norm ∥ċ∥H by The purpose of this note is to study the conditons on H for which the norm ∥ċ∥H is an absolute norm, a permutationally invariant norm, and a symmetric gauge function, respectively, if ∥ċ∥is a symmetric gauge function.