Sublinear function: Difference between revisions

Revision as of 02:03, 13 January 2022

Revision as of 02:27, 13 January 2022

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<li><math>p(0) = 0.</math>{{sfn|Narici|Beckenstein|2011|pp=120-121}}<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> <math>\blacksquare</math></ref></li>

<li><math>0 \leq p(x) + p(- x)</math> for every <math>x \in X.</math><ref group=proof><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> Consequently, at least one of <math>p(x)</math> and <math>p(- x)</math> must be nonnegative.</li>

<li><math>0 \leq \max \{p(x), p(- x)\}</math> for all <math>x \in X.</math>{{sfn|Narici|Beckenstein|2011|pp=120-121}}

* TheWhen <math>X</math> is 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 on <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=120-121}} </li>

* This implies, in particular, that at least one of <math>p(x)</math> and <math>p(- x)</math> is non-negative.</li>

<li><math>p(x) - p(y) \leq p(x - y)</math> for all <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></ref></li>

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===Associated seminorm===

If <math>p : X \to \R</math> is a real-valued sublinear function on a real vector space <math>X</math> (or if <math>X</math> is complex, then when it is considered as a real vector space) then the map <math>q(x) := \max \{p(x), p(- x)\}</math> defines a [[seminorm]] on the real vector space <math>X</math> called the '''seminorm associated with <math>p.</math>'''{{sfn|Narici|Beckenstein|2011|pp=120-121}}

A sublinear function <math>p</math> on a real or complex vector space is a [[#symmetric function|symmetric function]] if and only if <math>p = q</math> where <math>q(x) := \max \{p(x), p(- x)\}</math> as before.

More generally, if <math>p : X \to \R</math> is a real-valued sublinear function on a (real or complex) vector space <math>X</math> then

<math display=block>q(x) ~:=~ \sup_{|u|=1} p(u x) ~=~ \sup \{p(u x) : u \text{ is a unit scalar }\}</math>

defineswill define a [[seminorm]] on <math>X</math> if this supremum is always a real number (that is, never equal to <math>\infty</math>).

===Relation to linear functionals===