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Pseudo-norm

From Encyclopedia of Mathematics
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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=16614
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article