Counterexample for Mazur–Ulam theorem 
We know,  Mazur–Ulam theorem states that if $V$ and $W$ are normed spaces over $\mathbb{R}$ and the mapping
  $f\colon V\to W$
  is a surjective isometry, then $f$ is affine.

Can somebody say counterexample when $f\colon V\to W$ is no surjective then f is not affine.

I think that $f\colon \mathbb R\to L^1(\mathbb R)$ defined by
$$f(x)=\begin{cases}1_{[0,x]}&\text{if}\ x\geq 0\\ -1_{[0,-x]}&\text{if}\ x<0\end{cases}$$
is also a counterexemple.
