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)