Could you please help me in solving the following problem:
Let be the space of all such that , and are numbers, and be any norm on , and define the norm on to be . We want to show directly that there is such that for any , but without using the fact that all norms on a finite dimensional vector space are equivalent. Instead we want to use the fact that " A continuous mapping of a compact subset of a normed space into the real numbers assumes a maximum and a minimum at some points of ".
I think the continuous mapping in my problem will be the norm but which norms will be used and what is the compact set.