So, I am reading the following proposition in Neukirch's Algebraic Number Theory:
Proposition. Let $K$ be a complete field with respect to the valuation $|\:\: |$ and let V be an $n$-dimensional normed vector space over $K$. Then, for any basis $v_{1},...,v_{n}$ of $V$, the maximum norm
\begin{equation} ||(x_{1}v_{1}+\dots x_{n}v_{n}||=\max\{|x_{1}|,\dots , |x_{n}|\} \end{equation}
is equivalent to the given norm on $V$. In particular, $V$ is complete and the isomorphism
\begin{equation} K^{n}\to V,\qquad\qquad (x_{1},\:\dots , x_{n})\mapsto x_{1}v_{1}+\dots x_{n}v_{n} \end{equation}
is a homeomorphism.
Okay, so my question is: If this is proved and any norm in $V$ is indeed equivalent to the maximum norm in $V$, then why is the isomorphism above a homeomorphism?
I mean, he doesn't even mention which topology we have in $K^{n}$. Is there a canonical one that I should consider? Besides that, he says in the proof: "In fact, $||\: ||$ is transformed into the maximum norm on $K^{n}$", and that really got me lost on this.
Thanks in advance for any explanations.