Let $\mathfrak o$ be a Dedekind domain with field of fraction $K$, $\mathfrak p$ be one of its prime ideal, $K_{\mathfrak p}$ be the completion of $K$ at $\mathfrak p$, i.e. with respect to $| \,.|_\mathfrak p$, and $\mathfrak o_{\mathfrak p}$ be the valuation ring of $K_{\mathfrak p}$,
then for any $K$-vector space $V$:
$V\otimes_{\mathfrak o}\mathfrak o_{\mathfrak p} \approx V\otimes_K K_\mathfrak p$ by $v\otimes_{\mathfrak o}\lambda \mapsto v\otimes_K \lambda $ ($v\in V, \lambda \in \mathfrak o_{\mathfrak p}$)
I am not sure what type of isomorphism it is, I assume it's an $\mathfrak o$-module isomorphism.
The surjective part is easy, as for $K_{\mathfrak p}$ is $\mathfrak o_{\mathfrak p}$'s field of fraction, but why is it injective?