Let \((L,\|\cdot\|)\) and \((L,\|\cdot\|_{\ast})\) be normed spaces and let the norms \(\|\cdot\|\) and \(\|\cdot\|_{\ast}\) be equivalent. Then there exist the exact constants \(\alpha\) and \(\beta\) such that \(\alpha\|\cdot\|\leq\|\cdot\|_{\ast}\leq\beta\|\cdot\|.\) The functional \[ \rho\left(\|\cdot\|,\|\cdot\|_{\ast}\right) =\ln\frac{\beta}{\alpha} \] is a metric on the space \(K(L)\) of all equivalent norms. The space \(K(L)\) with this metric is a complete space. This functional is a distance between the normed spaces \((L,\|\cdot\|)\) and \((L,\|\cdot\|_{\ast}).\) The author proves that for any linear bounded operator \(A\) \[ e^{-\rho}=\inf_{A}\frac{\|A\|_{\ast}}{\|A\|},\;e^{\rho}=\sup_{A}\frac{\|A\|_{\ast}}{\|A\|}, \] where \(\rho=\rho\left(\|\cdot\|,\|\cdot\|_{\ast}\right).\) Some other properties of the proposed metric are discussed. The analogous results for the metric \[ \rho\left(\|\cdot\|,\|\cdot\|_{\ast}\right) =1-\frac{\alpha}{\beta} \] on the space \(K(L)\) were proposed by \textit{M. A. Krasnosel'skij, G. M. Vainikko, P. P. Zabrejko, Ya. B. Rutitskij} and \textit{V. Ya. Stetsenko} [``Approximate solution of operator equations''. Groningen (1972; Zbl 0231.41024)].