Editing Seminorm (section)

===Normability and seminormability===
{{See also|Normed space|Local boundedness#locally bounded topological vector space}}

A topological vector space (TVS) is said to be a {{em|{{visible anchor|seminormable space}}}} (respectively, a {{em|{{visible anchor|normable space}}}}) if its topology is induced by a single seminorm (resp. a single norm). 
A TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and [[T1 space|T<sub>1</sub>]] (because a TVS is Hausdorff if and only if it is a [[T1 space|T<sub>1</sub> space]]). 
A '''{{visible anchor|locally bounded topological vector space}}''' is a topological vector space that possesses a bounded neighborhood of the origin. 

Normability of [[topological vector space]]s is characterized by [[Kolmogorov's normability criterion]]. 
A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.{{sfn|Wilansky|2013|pp=50-51}} 
Thus a [[locally convex]] TVS is seminormable if and only if it has a non-empty bounded open set.{{sfn|Narici|Beckenstein|2011|pp=156-175}}
A TVS is normable if and only if it is a [[T1 space|T<sub>1</sub> space]] and admits a bounded convex neighborhood of the origin.

If <math>X</math> is a Hausdorff [[locally convex]] TVS then the following are equivalent: 
<ol>
<li><math>X</math> is normable.</li>
<li><math>X</math> is seminormable.</li>
<li><math>X</math> has a bounded neighborhood of the origin.</li>
<li>The [[strong dual]] <math>X^{\prime}_b</math> of <math>X</math> is normable.{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}</li>
<li>The strong dual <math>X^{\prime}_b</math> of <math>X</math> is [[Metrizable topological vector space|metrizable]].{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}</li>
</ol>
Furthermore, <math>X</math> is finite dimensional if and only if <math>X^{\prime}_{\sigma}</math> is normable (here <math>X^{\prime}_{\sigma}</math> denotes <math>X^{\prime}</math> endowed with the [[weak-* topology]]).

The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial (that is, 0-dimensional).{{sfn|Narici|Beckenstein|2011|pp=156–175}}