Let $\|\raisebox{1.72218pt}{.}\|$ be a norm of $V$ and $\mathscr{V}:=(V,\|\raisebox{1.72218pt}{.}\|)_{\leq 1}$. Then

$$\|v\|_{\mathscr{V}}=\inf\{|b|\mid\text{$b\in k^{\times}$ and $\|v\|\leq|b|$}\}.$$

Moreover, $\|\raisebox{1.72218pt}{.}\|\leq\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}}$ and $\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}}\leq|\alpha|\|\raisebox{1.72218pt}{.}\|$ for all $\alpha\in k^{\times}$ with $|\alpha|>1$.