Can a norm only be defined on vector spaces?

All examples I have come across so far deal with norm defined on a vector space. Can norm only be defined on vector spaces?

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That depends on your definition of "norm." – Qiaochu Yuan Dec 31 '10 at 9:54

For instance, a norm can be defined on an abelian group: see Section 2.4 of these notes.

It is perhaps less commonly done, but one can define a norm on any group: see Exercise 2.17 in the aforelinked lecture notes.

Of course there are many structures throughout mathematics that are called "norms", and you can make more if you want! But the above example is very closely related to the norm on a vector space: there is an induced metric, and so forth. In fact a norm on a vector space in the usual sense gives a norm on the underlying additive group $(V,+)$.

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The word "norm" is also used in connection with Euclidean domains (ring theory), although that Wikipedia page tries to avoid confusion by preferring instead "Euclidean function". Example: The degree of a univariate polynomial with coefficients over a field gives a norm for that kind of ring to be a Euclidean domain.

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There are other kinds of norms, like field norms and norms of ideals. See http://en.wikipedia.org/wiki/Norm#In_mathematics

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