Fix $\alpha\in(0,1)$. Let $(e_{1},\ldots,e_{r})$ be an $\alpha$-orthogonal basis of $V$ (cf. Proposition 1.3). We set

$$C_{1}=\max\{\|\phi(e_{1})\|^{\prime},\ldots,\|\phi(e_{r})\|^{\prime}\}\quad\text{and}\quad C_{2}=\min\{\|e_{1}\|,\ldots,\|e_{r}\|\}.$$

Then, for $v=a_{1}e_{1}+\cdots+a_{r}e_{r}\in V\setminus\{0\}$,

	$\displaystyle\frac{\|\phi(v)\|^{\prime}}{\|v\|}$	$\displaystyle\leq\frac{\max\{|a_{1}|\|\phi(e_{1})\|^{\prime},\ldots,|a_{r}|\|\phi(e_{r})\|^{\prime}\}}{\alpha\max\{|a_{1}|\|e_{1}\|,\ldots,|a_{r}|\|e_{r}\|\}}$
		$\displaystyle\leq\frac{\max\{|a_{1}|C_{1},\ldots,|a_{r}|C_{1}\}}{\alpha\max\{|a_{1}|C_{2},\ldots,|a_{r}|C_{2}\}}=\frac{C_{1}}{\alpha C_{2}},$

as desired. ∎