Why is the term 'norm' used instead of 'absolute value' in vector spaces?

[athrun200]

I saw some books and say that norm is the absolute value in vector.

If it also means absolute value, why don't we use absolute value |$\vec{v}$| instead we use ||$\vec{v}$||?

 

[Alchemista]

Absolute value is the usual norm for $\mathbb{R}$.
The euclidean norm is the usual norm for $\mathbb{R}^n$

While the euclidean norm is sometimes written using the same notation as absolute value, it is not the same thing. Furthermore, in the abstract a norm is not necessarily the euclidean norm.

http://en.wikipedia.org/wiki/Norm_(mathematics )

 

[HallsofIvy]

Strictly speaking, as Alchemista said, "absolute value" only applies to numbers. "norm" applies to any vector space, whether $R^n$ or more abstract, even infinite dimensional vector spaces. Of course, the set of real numbers can be thought of as a one-dimensional vector space and then the "usual norm" is, the absolute value.

Because of that, you will occaisionaly see the term "absolute value" used for the general norm but that is not very good terminology.