ScalingStacks

Proposition 2.15. Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ be a finite-dimensional ultrametrically normed vector space over $k$. Suppose that $(k,\lvert\mathord{\cdot}\rvert)$ is discretely valued. Then there exists an orthogonal basis for $(V,\lVert\mathord{\cdot}\rVert)$. ([BMPS, Proposition 2.5])