Here I proved the following result:
Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are equivalent (i.e., they induce the same topology) iff there are constants $c_1$ and $c_2$ such that $c_1p_1\leq p_2\leq c_2p_1$
And here Eric Wofsey showed, with a counterexample, that the proposition is false when the valuation is trivial (i.e. $|x|=1$, for each $x\in K^*$) and the space $X$ is infinite-dimensional.
Question: Is the proposition true when the valuation is trivial and $X$ is finite-dimensional?