19.2.1.6 Properties of normed spaces

Condition: Show the equivalence of the norms \( \|\cdot\|_{1},\|\cdot\|_{2} \) and \( \|\cdot\|_{\infty} \quad \) in the \( \quad \) finite-dimensional normed space \( X \), i.e. find positive constants \( c_{1} \) and \( c_{2} \) such that for any pair of norms \( \|\cdot\|^{\prime} \) and \( \|\cdot\|^{\prime \prime} \) the double inequality \( c_{1}\|x\|^{\prime} \leq\|x\|^{\prime \prime} \leq c_{2}\|x\|^{\prime} \) for all \( x \in X \).