Let $\|\raisebox{1.72218pt}{.}\|^{\prime}_{W_{k^{\prime}}}$ be the quotient norm of $\|\raisebox{1.72218pt}{.}\|_{V,k^{\prime}}$ with respect to the surjection $f_{k^{\prime}}:V_{k^{\prime}}\to W_{k^{\prime}}$. Let $e$ be an non-zero element of $W$. As $\|e\|_{W,k^{\prime}}=\|e\|_{W}$, it is sufficient to show that $\|e\|^{\prime}_{W_{k^{\prime}}}=\|e\|_{W}$. Note that

$$\{v\in V\mid f(v)=e\}\subseteq\{v^{\prime}\in V_{k^{\prime}}\mid f_{k^{\prime}}(v^{\prime})=e\},$$

so that we have $\|e\|_{W}\geq\|e\|^{\prime}_{W_{k^{\prime}}}$. Let us consider an inequality $\|e\|_{W}\leq\|e\|^{\prime}_{W_{k^{\prime}}}$. For $\epsilon>0$, let $(e_{1},\ldots,e_{r})$ be an $e^{-\epsilon}$-orthogonal basis of $V$ such that $(e_{2},\ldots,e_{r})$ forms a basis of $\operatorname{Ker}(f)$. Clearly we may assume that $f(e_{1})=e$. Then

	$\displaystyle\|e\|^{\prime}_{W_{k^{\prime}}}$	$\displaystyle=\inf\{\|e_{1}+a^{\prime}_{2}e_{2}+\cdots+a^{\prime}_{r}e_{r}\|_{V,k^{\prime}}\mid a^{\prime}_{2},\ldots,a^{\prime}_{r}\in k^{\prime}\}$
		$\displaystyle\geq\inf\{e^{-\epsilon}\max\{\|e_{1}\|,|a^{\prime}_{2}|^{\prime}\|e_{2}\|_{V},\ldots,|a^{\prime}_{r}|^{\prime}\|e_{r}\|_{V}\}\mid a^{\prime}_{2},\ldots,a^{\prime}_{r}\in k^{\prime}\}$
		$\displaystyle\geq e^{-\epsilon}\|e_{1}\|\geq e^{-\epsilon}\|e\|_{W}.$

Therefore, we have $\|e\|^{\prime}_{W_{k^{\prime}}}\geq\|e\|_{W}$ by taking $\epsilon\to 0$. ∎