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If <math>p</math> is a sublinear function on a real vector space <math>X</math> then there exists a linear functional <math>f</math> on <math>X</math> such that <math>f \leq p.</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}
If <math>X</math> is a real vector space, <math>f</math> is a linear functional on <math>X,</math> and <math>p</math> is a positive sublinear function on <math>X,</math> then <math>f \leq p</math> on <math>X</math> if and only if <math>f^{-1}(1) \cap \{
====Dominating a linear functional====
A real-valued function <math>f</math> defined on a subset of a real or complex vector space <math>X</math> is said to be {{em|dominated by}} a sublinear function <math>p</math> if <math>f(x) \leq p(x)</math> for every <math>x</math> that belongs to the ___domain of <math>f.</math>
If <math>f : X \to \R</math> is a real [[linear functional]] on <math>X</math> then{{sfn|Rudin|1991|pp=56-62}}{{sfn|Narici|Beckenstein|2011|pp=177-220}} <math>f</math> is dominated by <math>p</math> (that is, <math>f \leq p</math>) if and only if <math display=block>-p(-x) \leq f(x) \leq p(x) \quad \text{ for every } x \in X.</math>
Moreover, if <math>p</math> is a seminorm or some other {{em|symmetric map}} (which by definition means that <math>p(-x) = p(x)</math> holds for all <math>x</math>) then <math>f \leq p</math> if and only if <math>|f| \leq p.</math>
{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=177-220}}|math_statement=
If <math>p : X \to \R</math> be a sublinear function on a real vector space <math>X</math> and if <math>z \in X</math> then there exists a linear functional <math>f</math> on <math>X</math> that is dominated by <math>p</math> (that is, <math>f \leq p</math>) and satisfies <math>f(z) = p(z).</math>
Moreover, if <math>X</math> is a [[topological vector space]] and <math>p</math> is continuous at the origin then <math>f</math> is continuous.
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=== Continuity ===
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