# Pseudo-norm

A generalization of the concept of an absolute value or norm on a field, involving a weakening of one of the axioms: instead of the condition $w ( a \cdot b ) = w ( a) w ( b)$ only $w ( a \cdot b ) \leq w ( a) w ( b)$ is required. An example of a pseudo-norm: in the ring of all real-valued continuous functions $f$ defined on the segment $[ 0 , 1 ]$ a pseudo-norm which is not an absolute value is defined by the formula

$$p( f ) = \max _ {x \in [ 0 , 1 ] } | f ( x) | .$$

Every real finite-dimensional algebra can be given a pseudo-norm.

#### References

 [1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
How to Cite This Entry:
Pseudo-norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-norm&oldid=48348
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article