For $\alpha\in(0,1]$, let $(e_{1},\ldots,e_{r})$ be an $\alpha$-orthogonal basis of $V$ with respect to $\|\raisebox{1.72218pt}{.}\|$. Then $(e_{1}\otimes 1,\ldots,e_{r}\otimes 1)$ also yields an $\alpha$-orthogonal basis of $V_{k^{\prime}}$ with respect to $\|\raisebox{1.72218pt}{.}\|_{k^{\prime}}$.