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→Definitions: consistency |
→Definitions: Replaced erroneous inequality with correct equality Tags: Mobile edit Mobile app edit Android app edit App section source |
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Let <math>X</math> be a [[vector space]] over a field <math>\mathbb{K},</math> where <math>\mathbb{K}</math> is either the [[real number]]s <math>\Reals</math> or [[complex number]]s <math>\C.</math>
A real-valued function <math>p : X \to \mathbb{
<ol>
<li>''[[Positive homogeneity]]'''/'''[[Nonnegative homogeneity]]'':{{sfn|Schechter|1996|pp=313-315}} <math>p(r x) = r p(x)</math> for all real <math>r \geq 0</math> and all <math>x \in X.</math>
* This condition holds if and only if <math>p(r x)
<li>''[[Subadditivity]]'''/'''[[Triangle inequality]]'':{{sfn|Schechter|1996|pp=313-315}} <math>p(x + y) \leq p(x) + p(y)</math> for all <math>x, y \in X.</math>
* This subadditivity condition requires <math>p</math> to be real-valued.</li>
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