Editing Seminorm (section)

===Relationship to other norm-like concepts===

Let <math>p : X \to \R</math> be a non-negative function. The following are equivalent:
<ol>
<li><math>p</math> is a seminorm.</li>
<li><math>p</math> is a [[Convex function|convex]] [[F-seminorm|<math>F</math>-seminorm]].</li>
<li><math>p</math> is a convex balanced [[Metrizable topological vector space|''G''-seminorm]].{{sfn|Schechter|1996|p=691}}</li>
</ol>

If any of the above conditions hold, then the following are equivalent:
<ol>
<li><math>p</math> is a norm;</li>
<li><math>\{x \in X : p(x) < 1\}</math> does not contain a non-trivial vector subspace.{{sfn|Narici|Beckenstein|2011|p=149}}</li>
<li>There exists a [[Normed vector space|norm]] on <math>X,</math> with respect to which, <math>\{x \in X : p(x) < 1\}</math> is bounded.</li>
</ol>

If <math>p</math> is a sublinear function on a real vector space <math>X</math> then the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=177-220}} 
<ol>
<li><math>p</math> is a [[linear functional]];</li>
<li><math>p(x) + p(-x) \leq 0 \text{ for every } x \in X</math>;</li>
<li><math>p(x) + p(-x) = 0 \text{ for every } x \in X</math>;</li>
</ol>