Study of Some Balls in Space Rn

Abstract

In this work it presents different metrics in Rⁿ, not just the usual ones, known as module metrics, Euclidian, and maximum metrics, which are metric of the p metric case, with p in[1, ∞]. More precisely, it is a study about the sequence in function of p in which, for each fixed p, a different metric is obtained, for example, the Euclidean and module metrics are only particular cases for p = 1 and p = 2, respectively, since the maximum metric is treated as p = ∞. It will be shown that this sequence of functions dp converges to the maximum metric when p tends to infinity. In this way, the rules | · |p e | · |∞ and demonstrated that, in fact, they are metric for the set Rⁿ. To prove this result it is necessary to state and demonstrate three important inequalities: Elementary Inequality, H¨older Inequality and Minkowsky Inequality. Then, it is shown that when p tends to infinity, the norm | · |p converges to the standard k · k∞. At the end, it is concluded that all the metrics presented here are equivalent independent of the value p, in this way, one has the guarantee that whichever one of these metrics is obtained similar results on the space Rⁿ. These results are adaptations of similar results for infinite-dimensional spaces presented in Kreyszig (1978), with the last result being left as an exercise.