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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>
<ol>
<li>''[[Positive homogeneity]]'''/'''[[Nonnegative homogeneity]]'': <math>
* This
<li>''[[Subadditivity]]'''/'''[[Triangle inequality]]'': <math>
* This subadditivity condition requires <math>
</ol>
A function <math>
It is a ''{{em|{{visible anchor|symmetric function}}}}'' if <math>
Every subadditive symmetric function is necessarily nonnegative.<ref group=proof>Let <math>x \in X.</math> The triangle inequality and symmetry imply <math>p(0) = p(x + (- x)) \leq p(x) + p(-x) = p(x) + p(x) = 2 p(x).</math> Substituting <math>0</math> for <math>x</math> and then subtracting <math>p(0)</math> from both sides proves that <math>0 \leq p(0).</math> Thus <math>0 \leq p(0) \leq 2 p(x)</math> which implies <math>0 \leq p(x).</math> <math>\blacksquare</math></ref>
A sublinear function on a real vector space is [[#symmetric function|symmetric]] if and only if it is a [[seminorm]].
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\end{cases} \\
\end{alignat}</math>
is a sublinear function on <math>X := \R</math> and moreover,
If <math>p</math> and <math>q</math> are sublinear functions on a real vector space <math>X</math> then so is the map <math>x \mapsto \max \{p(x), q(x)\}.</math> More generally, if <math>\mathcal{P}</math> is any non-empty collection of sublinear functionals on a real vector space <math>X</math> and if for all <math>x \in X,</math> <math>q(x) := \sup \{p(x) : p \in \mathcal{P}\},</math> then <math>q</math> is a sublinear functional on <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=177-221}}
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==Properties==
Every sublinear function is a [[
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>
and in particular, 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>
<math display=block>p(x) - p(y) ~\leq~ p(x - y) \qquad \text{ for all } x, y \in X</math>
so if <math>p</math> is also [[#symmetric function|symmetric]] then the [[reverse triangle inequality]] will hold:
▲* When <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.{{sfn|Narici|Beckenstein|2011|pp=120-121}} </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>
===Associated seminorm===
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