I need help regarding the following two exercises:
a) Show that $(\mathbb R^2, d_2)$ and $(\mathbb R,d_1)$ $d_2,d_1$ being the respective euclidean metrics, are not isometric isomorphic, i.e. there is no bijective isometric map between those metric spaces.
b) Let d be the restriction of the euclidean metric of $\mathbb R$ on $(-1,1) \times (-1,1)$. Find a metric $d':\mathbb R \times \mathbb R\to \mathbb R$ such that the spaces $((-1,1),d)$ and $(\mathbb R,d')$ are isometric isomorphic.
For a) I think I have solution but I am not really sure:
Does it suffice to say that for every possible candidate $f$ for an isomorphism it can't be injective since it must be norm preserving, i.e.
$$||f(x)||_2 = ||x||_2, \quad ||f(-x)||_2 = ||-x||_2 = ||x||_2$$
with $||\cdot||_2$ being the respetive euclidean norm?
For b) I don't really know how to start.