EDB — 107

Exercises

1. [107] Prerequisites:[0PS]↺↻.Let \(X\) be a vector space, let \(𝜙,𝜓\) be two norms on it. Show that the topologies generated by \(𝜙\) and \(𝜓\) coincide, if and only if there exist \(0{\lt}a{\lt}b\) such that

   \begin{equation} ∀ x , ~ ~ a𝜓(x)≤𝜙(x)≤ b 𝜓(x)~ ~ .\label{eq:norme_ equiv} \end{equation}

   3

   (When the relation ?? holds, we will say that the norms are ”equivalent”).

   Solution 1

   [108]↺↻