Sublinear function: Difference between revisions

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Definitions: consistency
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Definitions: Replaced erroneous inequality with correct equality
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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{KR}</math> on <math>X</math> is called a ''{{em|{{visible anchor|sublinear function}}}}'' (or a ''{{em|{{visible anchor|sublinear functional|text=sublinear [[functional (mathematics)|functional]]}}}}'' if <math>\mathbb{K} = \Reals</math>), and also sometimes called a ''{{em|{{visible anchor|quasi-seminorm}}}}'' or a ''{{em|{{visible anchor|Banach functional}}}}'', if it has these two properties:{{sfn|Narici|Beckenstein|2011|pp=177-220}}
<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) \leq= r p(x)</math> for all positive real <math>r > 0</math> and all <math>x \in X.</math></li>
<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>