ScalingStacks

Remark 3.13. Given an NA norm $\left\lVert\cdot\right\rVert_{V}$ on the $K$-vector space $V=\mathbb{C}^{N+1}\otimes_{\mathbb{C}}K$, finding an orthogonal basis in general requires access to formal Laurent series. If we only use sections which are finite Laurent polynomials in $t$, then for any given $\epsilon>0$, we can find a $K$-basis $s_{0},\ldots s_{N}$ such that $\left\lVert a_{0}s_{0}+\ldots+a_{N}s_{N}\right\rVert_{V}^{\prime}=\max\{|a_{0}|\left\lVert s_{0}\right\rVert_{V},\ldots,|a_{N}|\left\lVert s_{N}\right\rVert_{V}\}$ satisfies

$$(1-\epsilon)\left\lVert\cdot\right\rVert_{V}^{\prime}\leq\left\lVert\cdot\right\rVert_{V}\leq\left\lVert\cdot\right\rVert_{V}^{\prime},$$

by [13, Prop. 1.3]. The upshot is that in the approximation theorem we may assume $s_{i}$ to be finite Laurent polynomials.