2.1.2. Orthogonal basis [00MJ]

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2.1.2. Orthogonal basis

Definition 2.12.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional normed vector space over $k$ . A basis $\{e_{i}\}_{i\in\{1,\dots,n\}}$ of $V$ is called orthogonal (with respect to $\lVert\mathord{\cdot}\rVert$ ) if

\forall(c_{1},\dots,c_{r})\in k^{n},\quad\Big\|\sum_{i\in\{1,\dots,n\}}c_{i}e_{i}\Big\|=\max_{i\in\{1,\dots,n\}}\lVert c_{i}e_{i}\rVert

Moreover, it is said to be orthonormal if in addition $\lVert e_{i}\rVert=1$ for all $i\in\{1,\dots,n\}$ .

Lemma 2.13.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional ultrametrically normed vector space over $k$ . If $\{v_{i}\}_{i\in\{1,\dots,n\}}$ is a finite set of elements of $V$ such that $\{\lVert v_{i}\rVert\}_{i\in\{1,\dots,n\}}$ are disctinct in $\mathbb{R}_{+}$ . Then $\lVert\sum_{i\in\{1,\dots,n\}}v_{i}\rVert=\max_{i\in\{1,\dots,n\}}\lVert v_{i}\rVert$ .

Proof.

If $n=2$ , this is clear from the ultra-metric inequality. For general $n$ an induction argument shows the equality. ∎

Corollary 2.14.

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ be a finite-dimensional ultrametrically normed vector space over $k$ . Suppose that $(k,|\cdot|)$ is discretely valued. If $\{e_{i}\}_{i\in\{1,\dots,n\}}$ is a basis of $V$ such that $\{\log\lVert e_{i}\rVert\}_{i\in\{1,\dots,n\}}$ are $\mathbb{Q}$ -independent in $\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$ , then $\{e_{i}\}_{i\in\{1,\dots,n\}}$ is an orthogonal basis.

Proof.

For any $f=(f_{1},\dots,f_{n})\in(k^{\times})^{n}$ , the numbers $\{\lvert f_{i}\rvert\lVert e_{i}\rVert\}_{i\in\{1,\dots,r\}}$ are distinct, otherwise there exist $i,j\in\{1,\dots,n\},i\neq j$ such that

\log\lVert e_{i}\rVert-\log\lVert e_{j}\rVert=\log\Big|\frac{f_{i}}{f_{j}}\Big|\in\log\lvert k^{\times}\rvert

which contradicts the assumption of $\mathbb{Q}$ -independence. Hence

\Big\|\sum_{i\in\{1,\dots,n\}}f_{i}e_{i}\Big\|=\max_{0\leq i\leq n}\lVert f_{i}e_{i}\rVert

by Lemma 2.13. ∎

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])