These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points (maybe a reference?).
If we have a vector space $V$ with a norm $||\cdot||$ then it implies that $(V,d)$ is a metric space with metric $d(x,y)=||x-y||$. In turn, this gives access to huge parts of analysis (by using the metric to define open/closed sets, limits, continuity, etc.). But apart from inducing metrics what else are norms good for?
If applicable - in how many of these uses could $d(x,0)$, where $0$ denotes additive identity of $V$, replace $||\cdot||$? If $||\cdot||$ cannot be replaced with $d(x,0)$, why? Is it maybe that homogeniety of $||\cdot||$ is particularly important?