Let $(e_{1},\ldots,e_{r})$ be an $\alpha$-orthogonal basis of $V$ such that $W=ke_{1}+\cdots+ke_{l}$ (cf. Proposition 1.3). We define $\varphi\in V^{\vee}$ to be

$$\varphi(a_{1}e_{1}+\cdots+a_{r}e_{r}):=\psi(a_{1}e_{1}+\cdots+a_{l}e_{l})$$

for $a_{1},\ldots,a_{r}\in k$. Then $\left.{\varphi}\right|_{{W}}=\psi$. Moreover, note that

$$\alpha\|a_{1}e_{1}+\cdots+a_{l}e_{l}\|\leq\alpha\max\{|a_{1}|\|e_{1}\|,\ldots,|a_{l}|\|e_{l}\|\}\\ \leq\alpha\max\{|a_{1}|\|e_{1}\|,\ldots,|a_{r}|\|e_{r}\|\}\leq\|a_{1}e_{1}+\cdots+a_{r}e_{r}\|,$$

so that

$$\frac{|\varphi(a_{1}e_{1}+\cdots+a_{r}e_{r})|}{\|a_{1}e_{1}+\cdots+a_{r}e_{r}\|}\leq\alpha^{-1}\frac{|\psi(a_{1}e_{1}+\cdots+a_{l}e_{l})|}{\|a_{1}e_{1}+\cdots+a_{l}e_{l}\|}\leq\alpha^{-1}\|\psi\|^{\vee}$$

for all $a_{1},\ldots,a_{r}\in k$ with $(a_{1},\ldots,a_{l})\not=(0,\ldots,0)$. Thus the assertion follows. ∎