Is the quotient space of a vector space with a dense subspace isomorphic to the underlying field?

Question

Let $\mathbb{K}$ be a field, $V$ be a $\mathbb{K}$-Banach space and $U<V$ be a dense subspace (with $\overline{U}=V$). I have heard, that: $$V/U\cong\mathbb{K}.$$ Is this result even correct and if not, what is a simple counterexample? If it is, how do you prove it? Unfortunately, it looks a lot like it can simply be done with the fundamental theorem on homomorphisms, but that is not the case. There is no surjective function $f\colon V\twoheadrightarrow\mathbb{K}$ with $\ker(f)=U\Rightarrow f\vert_U=0$ since the latter condition combined with $U$ being dense implies $f=0$.