Proof. [00LY]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Proof.

By the orthogonality of the basis $\{T_{i}\}_{i\in\{0,\dots,d\}}$ , we can compute

\begin{split}\lvert T_{i}\rvert_{\phi}(x(\boldsymbol{r}))&=\inf_{\begin{subarray}{c}(\sum_{m\in\{0,\dots,d\}}f_{m}\cdot T_{m})(x(\boldsymbol{r}))=(T_{i})(x(\boldsymbol{r}))\\ (f_{0},\dots,f_{d})\in k^{d+1}\end{subarray}}\Big\lVert\sum_{m\in\{0,\dots,d\}}f_{m}\cdot T_{m}\Big\rVert_{\phi}\\ &=\inf\limits_{\sum_{m\in\{0,\dots,d\}}f_{m}\kappa_{m}=\kappa_{i}}\max_{m\in\{0,\dots,d\}}\Big\{\lVert f_{m}\cdot T_{m}\rVert_{\phi}\Big\}\\ &=\inf\limits_{\sum_{m\in\{0,\dots,d\}}f_{m}\kappa_{m}=\kappa_{i}}\max_{m\in\{0,\dots,d\}}\Big\{\lvert f_{m}\rvert\lvert\kappa_{m}\rvert\Big\}\\ &=\inf\limits_{\sum_{m\in\{0,\dots,d\}}f_{m}(\kappa_{m}/\kappa_{i})=1}\max_{m\in\{0,\dots,d\}}\Big\{\lvert f_{m}\rvert\lvert\kappa_{m}/\kappa_{i}\rvert\cdot\lvert\kappa_{i}\rvert\Big\}\\ &=\lvert\kappa_{i}\rvert=r_{i}.\end{split}

The last equality is obtained by Lemma 3.13. ∎