How does one define an inner product on the space $V=\mathbb{Q}_p^n$?

Question

In J H H Chalk's paper "Algebraic Lattices", he defines the notion of a lattice in ultrametric space as the following:

Given a non-archimedean field $K$ with ring of integers $\mathcal{O}_K$ and completion $\tilde{K}$,

1. The integer lattice is defined $\Lambda_0 = \mathcal{O}_K^n \subset V= \tilde{K}^n$,

2. A lattice $\Lambda$ in $V$ is the image of $\Lambda_0$ under an invertible $\tilde{K}$-linear function $\lambda$ that sends $V$ to itself. The determinant $d(\lambda) = |\det{\lambda}|$,

3. $\|\mathbf{x} - \mathbf{y}\|=\max_{1\leq i \leq n}|x_i - y_i|$ for all $\mathbf{x},\mathbf{y} \in V$ with entries $x_i, y_i$.

Define $K = \mathcal{Q}_p$ so its ring of integers is the $p$-adic integers. How can we define an inner product on this space that satisfies the properties of an inner product, e.g. to find the dual of a lattice?

Answer 1

On the complete metric space $\mathbb{Q}_p^n$ we introduce the inner product as usual: $$ (x,y)=x_1y_1+\cdots +x_ny_n $$ for $x=(x_1,\ldots ,x_n)$ and $y=(y_1,\ldots ,y_n)$. This satisfies the inequality $$ |(x,y)|_p\le |x|_p\cdot |y|_p. $$

Edit: As $K=\mathbb{Q}_p$ is not ordered, the usual third axiom $(x,x)> 0$ for $x\neq 0$ is commonly replaced by the axiom $(x,x)\neq 0$ whenever $x\neq 0$.