My question is if any isometry $f:V\to W$ between real normed spaces sends lines to lines. I've seen several questions/answers about this but only in euclidean spaces.
So I thought it was false on general (real) normed spaces. However I found this theorem of Mazur-Ulam: any surjective isometry $f:V\to W$ is an affine map, hence it maps lines to lines.
But if my isometry is not surjective, would this still apply? I think that considering the image space $f(V)$ it would be the same, because $f:V\to f(V)$ is affine and any line $L$ would be sent to a line $f(L)$ in $f(V)$ which is also a line in $W$.
Is this correct?