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Seminorm
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==Generalizations== The concept of {{em|norm}} in [[composition algebra]]s does {{em|not}} share the usual properties of a norm. A composition algebra <math>(A, *, N)</math> consists of an [[algebra over a field]] <math>A,</math> an [[Involution (mathematics)|involution]] <math>\,*,</math> and a [[quadratic form]] <math>N,</math> which is called the "norm". In several cases <math>N</math> is an [[isotropic quadratic form]] so that <math>A</math> has at least one [[null vector]], contrary to the separation of points required for the usual norm discussed in this article. An {{em|ultraseminorm}} or a {{em|non-Archimedean seminorm}} is a seminorm <math>p : X \to \R</math> that also satisfies <math>p(x + y) \leq \max \{p(x), p(y)\} \text{ for all } x, y \in X.</math> '''Weakening subadditivity: Quasi-seminorms''' A map <math>p : X \to \R</math> is called a {{em|[[Quasinorm|quasi-seminorm]]}} if it is (absolutely) homogeneous and there exists some <math>b \leq 1</math> such that <math>p(x + y) \leq b p(p(x) + p(y)) \text{ for all } x, y \in X.</math> The smallest value of <math>b</math> for which this holds is called the {{em|multiplier of <math>p.</math>}} A quasi-seminorm that separates points is called a {{em|quasi-norm}} on <math>X.</math> '''Weakening homogeneity - <math>k</math>-seminorms''' A map <math>p : X \to \R</math> is called a {{em|<math>k</math>-seminorm}} if it is subadditive and there exists a <math>k</math> such that <math>0 < k \leq 1</math> and for all <math>x \in X</math> and scalars <math>s,</math><math display="block">p(s x) = |s|^k p(x)</math> A <math>k</math>-seminorm that separates points is called a {{em|<math>k</math>-norm}} on <math>X.</math> We have the following relationship between quasi-seminorms and <math>k</math>-seminorms: {{block indent | em = 1.5 | text = Suppose that <math>q</math> is a quasi-seminorm on a vector space <math>X</math> with multiplier <math>b.</math> If <math>0 < \sqrt{k} < \log_2 b</math> then there exists <math>k</math>-seminorm <math>p</math> on <math>X</math> equivalent to <math>q.</math>}}
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