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Every [[Norm (mathematics)|norm]], [[seminorm]], and real linear functional is a sublinear function.
The [[identity function]] <math>\R \to \R</math> on <math>X := \R</math> is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation <math>x \mapsto -x.</math>{{sfn|Narici|Beckenstein|2011|pp=177-221}}
More generally, for any real <math>a \
<math display=block>\begin{alignat}{4}
S_{a,b} :\;&& \R &&\;\to \;& \R \\[0.3ex]
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\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> and <math>-f(-1) \
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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