Editing Seminorm (section)

===Continuity of linear maps===

If <math>F : (X, p) \to (Y, q)</math> is a map between seminormed spaces then let{{sfn|Wilansky|2013|pp=21-26}}
<math display="block">\|F\|_{p,q} := \sup \{q(F(x)) : p(x) \leq 1, x \in X\}.</math>

If <math>F : (X, p) \to (Y, q)</math> is a linear map between seminormed spaces then the following are equivalent:
<ol>
<li><math>F</math> is continuous;</li>
<li><math>\|F\|_{p,q} < \infty</math>;{{sfn|Wilansky|2013|pp=21-26}}</li>
<li>There exists a real <math>K \geq 0</math> such that <math>p \leq K q</math>;{{sfn|Wilansky|2013|pp=21-26}}
* In this case, <math>\|F\|_{p,q} \leq K.</math></li>
</ol>
If <math>F</math> is continuous then <math>q(F(x)) \leq \|F\|_{p,q} p(x)</math> for all <math>x \in X.</math>{{sfn|Wilansky|2013|pp=21-26}}

The space of all continuous linear maps <math>F : (X, p) \to (Y, q)</math> between seminormed spaces is itself a seminormed space under the seminorm <math>\|F\|_{p,q}.</math> 
This seminorm is a norm if <math>q</math> is a norm.{{sfn|Wilansky|2013|pp=21-26}}