EDB — 109

Exercises

1. [109] We want to show that ”the norms in \(ℝ^ n\) are all equivalent.”

   Let \(\| x\| =\sqrt{∑_{i=1}^ n x_ i^ 2}\) be the Euclidean norm. Let \(𝜙:ℝ^ n→[0,∞)\) 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 \(0{\lt}a{\lt}b\) such that

   \begin{equation} ∀ x , ~ ~ a\| x\| ≤𝜙(x)≤ b\| x\| ~ ~ .\label{eq:Rn_ norma_ equiv} \end{equation}

   4

   Solution 1

   [10B]↺↻