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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?).

  1. 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?

  2. 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?

Thanks.

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When one studies topological vector spaces, often one topologizes by declaring the open sets of the identity, then asking that open sets about other points are obtained by shifting the opens about the origin. The norm does this for free. The norm also give a metric structure that is useful and compatible homogenius. So it is not just the homogeniety of the topology, but of the metric. –  Baby Dragon Mar 28 '13 at 18:54

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