2.1. Seminormed vector spaces [00MH]

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2.1. Seminormed vector spaces

2.1.1. Basic constructions

Let $V$ be a vector space over $k$ . By seminorm on $V$ , we refer to a map $\lVert\mathord{\cdot}\rVert:V\rightarrow\mathbb{R}_{\geq 0}$ such that $\lVert ax\rVert=\lvert a\rvert\cdot\lVert x\rVert$ for any $(a,x)\in k\times V$ and that $\lVert x+y\rVert\leqslant\lVert x\rVert+\lVert y\rVert$ for any $(x,y)\in V\times V$ . The couple $(V,\lVert\mathord{\cdot}\rVert)$ is called a seminormed vector space over $k$ . If in addition $\lVert\mathord{\cdot}\rVert$ takes positive values on $V\setminus 0$ , we say that $\lVert\mathord{\cdot}\rVert$ is a norm on $V$ and that $(V,\lVert\mathord{\cdot}\rVert)$ is a normed vector space. Denote by $\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ the inverse image of $\{0\}$ by $\lVert\mathord{\cdot}\rVert$ . It can be shown that $\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ is a closed vector subspace of $V$ , called the null space of $\lVert\mathord{\cdot}\rVert$ . Note that there exists a unique norm on $V/\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ , the composition of which with the projection map $V\rightarrow V/\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ identifies with the seminorm $\lVert\mathord{\cdot}\rVert$ . Call this norm the induced norm of $\lVert\mathord{\cdot}\rVert$ .

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a seminormed vector space over $k$ . If the strong triangle inequality holds for the seminorm $\lVert\mathord{\cdot}\rVert$ , namely $\lVert x+y\rVert\leqslant\max(\lVert x\rVert,\lVert y\rVert)$ for any $(x,y)\in V\times V$ , we say that the seminorm is ultrametric. Note that if a seminorm is ultra-metric, then the inequality becomes an equality whenever $\lVert x\rVert\neq\lVert y\rVert$ .

We say that a seminormed (resp. normed) vector space $(V,\lVert\mathord{\cdot}\rVert)$ is complete, or $\lVert\mathord{\cdot}\rVert$ is a complete seminorm (resp. complete norm) on $V$ , if any Cauchy sequence in $V$ with respect to the seminorm $\lVert\mathord{\cdot}\rVert$ admits a limit. A complete normed vector space over $k$ is called a Banach space over $k$ . Any finite-dimensional normed space $(V,\lVert\mathord{\cdot}\rVert)$ is complete. ([Bou, 1.2.3 Theorem 2])

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ be a seminormed vector space over $k$ . Let $\widetilde{V}_{c}$ be the vector space of all Cauchy sequences in $V$ with respect to $\lVert\mathord{\cdot}\rVert_{V}$ . We define a seminorm $\lVert\mathord{\cdot}\rVert_{c}$ on $\widetilde{V}_{c}$ which sends any Cauchy sequence $\{v_{i}\}_{i\in\mathbb{N}}$ to $\lim_{i\rightarrow+\infty}\lVert v_{i}\rVert_{V}$ . Denote by $V_{c}$ the quotient vector space $V_{c}/\mathfrak{n}(\lVert\mathord{\cdot}\rVert_{c})$ . Then the vector space $V_{c}$ equipped with the norm induced by $\lVert\mathord{\cdot}\rVert_{c}$ forms a Banach space over $k$ , called the separated completion of $(V,\lVert\mathord{\cdot}\rVert)$ . Tautologically it can be shown that this Banach space is canonically isomorphic to the completion of $V/\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ equipped with the quotient norm induced by the seminorm $\lVert\mathord{\cdot}\rVert$ .

Definition 2.1.

Let $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ be seminorms on $V$ . We say that $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ are equivalent if there exist two constants $C_{1}>0$ and $C_{2}>0$ such that $C_{1}\lVert\mathord{\cdot}\rVert_{1}\leq\lVert\mathord{\cdot}\rVert_{2}\leq C_{2}\lVert\mathord{\cdot}\rVert_{1}$ . Note that this condition holds if and only if the seminorms $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ induce the same topology on the vector space $V$ ([Bou, Corollaire I.3.3.1])(note that the absolute value $\lvert\mathord{\cdot}\rvert$ is supposed to be non-trivial).

Definition 2.2.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a seminormed vector space over $k$ . If $W$ is a vector subspace of $V$ , then map $(x\in W)\mapsto\lVert x\rVert$ defines a seminorm on $W$ , called the restriction of $\lVert\mathord{\cdot}\rVert$ on $W$ . If $Q$ is a quotient vector space of $V$ and $\pi:V\rightarrow Q$ is the quotient map, then the map $(q\in Q)\mapsto\inf_{x\in\pi^{-1}(\{q\})}\lVert x\rVert$ defines a seminorm on $Q$ , called the quotient of $\lVert\mathord{\cdot}\rVert$ on $Q$ .

Definition 2.3.

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ and $(W,\lVert\mathord{\cdot}\rVert_{W})$ be seminormed vector spaces over $k$ , and $f:V\rightarrow W$ be a $k$ -linear map. We say that $f$ is bounded if there exists a constant $C>0$ such that $\lVert f(x)\rVert_{W}\leq C\lVert x\rVert_{V}$ for any $x\in V$ . Note that this condition holds if and only if $f$ is continuous with respect to the topologies on $V$ and $W$ induced by the seminorms $\lVert\mathord{\cdot}\rVert_{V}$ and $\lVert\mathord{\cdot}\rVert_{W}$ respectively. We say that $f$ is admissible if it is bounded and if on the image of $f$ , the quotient seminorm of $\lVert\mathord{\cdot}\rVert_{V}$ and the restriction of $\lVert\mathord{\cdot}\rVert_{W}$ are equivalent.

We recall below several fundamental results in functional analysis and refer to [Bou, Theorem 1.3.3.1, Corollary 1.3.3.1, 1.3.3.2, 1.3.3.5] for more details.

Theorem 2.4.

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ and $(W,\lVert\mathord{\cdot}\rVert_{W})$ be Banach spaces over $k$ , and $f:V\rightarrow W$ be a $k$ -linear map.

(1)

The $k$ -linear map $f$ is bounded if and only if its graph in $V\times W$ is closed under the product topology.
(2)

Assume that $f$ is bounded and surjective, then $f$ is an open map. In particular, the quotient norm of $\lVert\mathord{\cdot}\rVert_{V}$ on $W$ is equivalent to $\lVert\mathord{\cdot}\rVert_{W}$ .
(3)

Assume that $f$ is bounded and injective, then $f(V)$ is closed in $W$ .

Theorem 2.5.

Let $V$ be a vector space over $k$ and $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ be complete norms on $V$ . If there exists $C>0$ such that $\lVert\mathord{\cdot}\rVert_{2}\leq C\lVert\mathord{\cdot}\rVert_{1}$ , then the norms $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ are equivalent.

Using this norm equivalence theorem for Banach spaces over $k$ , we have immediately the following

Corollary 2.6.

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ and $(W,\lVert\mathord{\cdot}\rVert_{W})$ be Banach spaces over $k$ , and $f:V\rightarrow W$ be a bounded $k$ -linear map with closed image. Then $f$ is admissible.

Definition 2.7.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional normed vector space. The dual norm of $\lVert\mathord{\cdot}\rVert^{\vee}$ on the dual vector space $V^{\vee}$ is defined by

\forall\ell\in V^{\vee},\ \lVert\ell\rVert^{\vee}:=\sup_{v\in V\setminus\{0\}}\frac{\lvert\ell(v)\rvert}{\lVert v\rVert}.

Remark 2.8.

The norm $\lVert\mathord{\cdot}\rVert^{\vee}$ is ultrametric, and $\lVert\mathord{\cdot}\rVert^{\vee\vee}=\lVert\mathord{\cdot}\rVert$ if and only if $\lVert\mathord{\cdot}\rVert$ is ultrametric. ([CMor18, Section 2.2.3])

Definition 2.9.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a normed vector space. Let $(k^{\prime},\lvert\mathord{\cdot}\rvert^{\prime})$ be a complete valued field extension of $(k,\lvert\mathord{\cdot}\rvert)$ . Set $V_{k^{\prime}}$ to be $V\otimes_{k}k^{\prime}$ , which can be identified with $\mathrm{Hom}_{k}(\mathrm{Hom}_{k}(V,k),k^{\prime})$ . The norm

\forall v^{\prime}\in V_{k^{\prime}},\ \lVert v^{\prime}\rVert_{k^{\prime}}:=\sup\Big\{\frac{\lvert(\ell\otimes 1)(v^{\prime})\rvert^{\prime}}{\lVert\ell\rVert^{\vee}},\ \ell\in V^{\vee}\Big\}

defined via this identification is called the scalar extension of $\lVert\mathord{\cdot}\rVert$ .

Remark 2.10.

If $\lVert\mathord{\cdot}\rVert$ is ultrametric, then $\lVert\mathord{\cdot}\rVert_{k^{\prime}}$ is the largest ultrametric norm on $V_{k^{\prime}}$ extending $\lVert\mathord{\cdot}\rVert$ . ([CMor18, Definition 2.4])

Lemma 2.11.

Let $f:V\rightarrow W$ be a surjective $k$ -linear map of finite-dimensional vector spaces, with $\mathrm{dim}_{k}W=1$ . Let $\lVert\mathord{\cdot}\rVert_{V}$ be a norm on $V$ and let $\lVert\mathord{\cdot}\rVert_{W}$ be its quotient norm for $f$ . Then the norm $\lVert\mathord{\cdot}\rVert_{W,k^{\prime}}$ identifies with the quotient norm of $\lVert\mathord{\cdot}\rVert_{V,k^{\prime}}$ induced by the surjective $k^{\prime}$ -linear map $f\otimes\mathrm{id}_{k^{\prime}}:V_{k^{\prime}}\rightarrow W_{k^{\prime}}$ . ([CMor18, Lemma 2.5])

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])