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Difference between revisions of "Pseudo-norm"

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A generalization of the concept of an [[Absolute value|absolute value]] or [[Norm on a field|norm on a field]], involving a weakening of one of the axioms: instead of the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075750/p0757501.png" /> only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075750/p0757502.png" /> is required. An example of a pseudo-norm: in the ring of all real-valued continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075750/p0757503.png" /> defined on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075750/p0757504.png" /> a pseudo-norm which is not an absolute value is defined by the formula
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A generalization of the concept of an [[Absolute value|absolute value]] or [[Norm on a field|norm on a field]], involving a weakening of one of the axioms: instead of the condition  $  w ( a \cdot b ) = w ( a) w ( b) $
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only  $  w ( a \cdot b ) \leq  w ( a) w ( b) $
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is required. An example of a pseudo-norm: in the ring of all real-valued continuous functions  $  f $
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defined on the segment  $  [ 0 , 1 ] $
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a pseudo-norm which is not an absolute value is defined by the formula
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$$
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p( f  )  = \max _ {x \in [ 0 , 1 ] }  | f ( x) | .
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$$
  
 
Every real finite-dimensional algebra can be given a pseudo-norm.
 
Every real finite-dimensional algebra can be given a pseudo-norm.

Latest revision as of 08:08, 6 June 2020


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
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