Sublinear function: Difference between revisions

Revision as of 04:51, 20 November 2022

Revision as of 00:44, 17 January 2023

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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}}

Subadditivity of <math>p : X \to \R</math> guarantees for all vectors <math> x, y \in X,</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}<ref group=proof><math>p(x) = p(y + (x - y)) \leq p(y) + p(x - y),</math> which happens if and only if <math>p(x) - p(y) \leq p(x - y).</math> <math>\blacksquare</math> Substituting <math>y := -x</math> and gives <math>p(x) - p(-x) \leq p(x - (-x)) = p(x + x) \leq p(x) + p(x),</math> which implies <math>- p(-x) \leq p(x)</math> (positive homogeneity is not needed; the triangle inequality suffices). <math>\blacksquare</math></ref>

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

<math display=block>- p(x) ~\leq~ p(- x),</math>

so if <math>p</math> is also [[#symmetric function|symmetric]] then the [[reverse triangle inequality]] will hold:

so if <math display=block>|p(x)</math> -is p(y)also [[#symmetric function|symmetric]] ~\leq~then p(xthe -[[reverse y)triangle \qquadinequality]] \text{will hold for all }vectors <math> x, y \in X.,</math>

<math display=block>|p(x) - p(y)| ~\leq~ p(x - y).</math>

{{Math theorem