Sublinear function: Difference between revisions

Revision as of 05:31, 29 January 2022

Revision as of 05:31, 29 January 2022

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If <math>p : X \to \R</math> is a sublinear function on a vector space <math>X</math> then<ref group=proof>If <math>x \in X</math> and <math>r := 0</math> then nonnegative homogeneity implies that <math>p(0) = p(r x) = r p(x) = 0 p(x) = 0.</math> Consequently, <math>0 = p(0) = p(x + (-x)) \leq p(x) + p(-x),</math> which is only possible if <math>0 \leq \max \{p(x), p(- x)\}.</math> <math>\blacksquare</math></ref>{{sfn|Narici|Beckenstein|2011|pp=120-121}}

<math display=block>p(0) ~=~ 0 ~\leq~ p(x) + p(- x) \qquad \text{ for every } x \in X,</math>

andwhich inimplies particular,that at least one of <math>p(x)</math> and <math>p(- x)</math> must be nonnegative; that is,{{sfn|Narici|Beckenstein|2011|pp=120-121}}

<math display=block>0 ~\leq~ \max \{p(x), p(- x)\} \qquad \text{ for every } x \in X.</math>

Moreover, when <math>p : X \to \R</math> is a sublinear function on a real vector space then the map <math>q : X \to \R</math> defined by <math>q(x) := \max \{p(x), p(- x)\}</math> is a seminorm.{{sfn|Narici|Beckenstein|2011|pp=120-121}}