Let \(\textbf{K} \cdot \textbf{K}_1\) and \(\textbf{K} \cdot \textbf{K}_2\) be two norms defined on a linear space X. \(\textbf{K} \cdot \textbf{K}_1\) and \(\textbf{K} \cdot \textbf{K}_2\) are equivalent if there exists constants \(a, b > 0\) such that \(\hspace{1.0cm}\) for all \(x \in X\).
\(a \textbf{K} \times \textbf{K}_1 \geq 0\)
\(a \textbf{K} \times \textbf{K}_1 \geq \textbf{K} \times \textbf{K}_2\)
\(a \textbf{K} \times \textbf{K}_1 \leq \textbf{K} \times \textbf{K}_2\)
—>> \(a \textbf{K} \times \textbf{K}_1 \leq \textbf{K} \times \textbf{K}_2 \leq b\textbf{K} \times \textbf{K}_1 \)
