EDB — 109

Exercises

1. [109] We want to show that ”the norms in ${ℝ}^n$ are all equivalent.”

   Let $\Vert x\Vert =\sqrt{{\sum }_{i=1}^n{x}_i^2}$ be the Euclidean norm. Let $𝜙:{ℝ}^n\to [0,\infty )$ be a norm: it can be shown that $𝜙$ is a convex function, see [0ZX]↺↻; and therefore $𝜙$ is a continuous function, see [186]↺↻. Use this fact to prove that there exist $0<a<b$ such that

   $$\begin{array}{}\text{(1)}& \forall x,a\Vert x\Vert \le 𝜙(x)\le b\Vert x\Vert .\end{array}$$

   4

   Solution 1

   [10B]↺↻