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A real-valued function <math>f : X \to \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>f(
* This statement holds if and only if <math>f(r x) \leq r f(x)</math> for all positive real <math>r > 0</math> and all <math>x \in X.</math></li>
<li>''[[Subadditivity]]'''/'''[[Triangle inequality]]'': <math>f(x + y) \leq f(x) + f(y)</math> for all <math>x, y \in X.</math>
* This subadditivity condition requires <math>f</math> to be real-valued.</li>
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