Sublinear function: Difference between revisions

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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>\R</math> or [[complex number]]s <math>\C.</math>

A real-valued function <math>p : X \to \mathbb{K}R</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} = \R</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]]'': <math>p(r x) = r p(x)</math> for all real <math>r \geq 0</math> and all <math>x \in X.</math>