# Pseudo-norm

From Encyclopedia of Mathematics

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