I know that not all metrics are induced by norm, only those metrics satisfy the homogeneity and translation invariant.
1. Now, Can a normed space equipped with metric not be induced by the norm?
2. If it can, consider this metric if $x\neq y$ $$d(x,y):=\max \{\lVert x-y\rVert_{2},1 \}$$ $$d(x,y):=0\quad x=y$$ with the normed space, my motivation of this metric is bounded below to make sure that there can not have a Cauchy sequence in the space (except for the trivial constant sequence). In such case, is the space vacuously complete by no non-constant Cauchy sequence exists, so that every Cauchy sequence convergence in the space?
Please help me out with these two questions