Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|cite|improve this question
That depends on your definition of "norm." –  Qiaochu Yuan Dec 31 '10 at 9:54

3 Answers 3

up vote 13 down vote accepted

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 $(V,+)$.

share|cite|improve this answer

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.

share|cite|improve this answer

There are other kinds of norms, like field norms and norms of ideals. See

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.