Sublinear function: Difference between revisions

Revision as of 00:47, 28 January 2022

Revision as of 02:07, 28 January 2022

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==Examples and sufficient conditions==

Every [[seminorm]] and [[Norm (mathematics)|norm]], is a sublinear function[[seminorm]], and every real linear functional is a sublinear function. The converses are not true in general.

The linear[[identity functionalfunction]] <math>x\R \mapstoto -x\R</math> on <math>X := \R</math> is an example of a sublinear function (in fact, it is even a linear functional) that is notneither positive andnor a seminorm; the same is nottrue aof seminormthis map's negation <math>x \mapsto -x.</math>{{sfn|Narici|Beckenstein|2011|pp=177-221}}

More generally, for any real <math>a \geq b,</math> the map

<math display=block>\begin{alignat}{4}

S_{a,b} :\;&& \R &&\;\to \;& \R \\[0.3ex]

&& x &&\;\mapsto\;&

\begin{cases}

a x & \text{ if } x \leq 0 \\

b x & \text{ if } x \geq 0 \\

\end{cases} \\

\end{alignat}</math>

is a sublinear function on <math>X := \R</math> and moreover, if <math>f : \R \to \R</math> is any sublinear function then <math>f = S_{f(1), -f(-1)}.</math>

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}}

The linear functional <math>x \mapsto -x</math> on <math>X = \R</math> is a sublinear functional that is not positive and is not a seminorm.{{sfn|Narici|Beckenstein|2011|pp=177-221}}

==Properties==