Suppose $E$ be a $\mathbb R-$vector space that is isometrically isomorphic to $\mathbb R^n$ via the map $f:\, E\to\mathbb R^n$, which means $f$ is bijective and
\begin{align}
d_E(x,y)&=d_{\mathbb R^n}\big(f(x),f(y)\big)
\\ f(x+y)&=f(x)+f(y)
\end{align}
for all $x,y\in E$.
My question is if $f$ is a diffeomorphism ? I have no clue for this confusion, may anyone provide me with some explaination, some interesting idea ? Thanks