Editing Seminorm (section)

===Topological properties===

<ul>
<li>If <math>X</math> is a TVS and <math>p</math> is a continuous seminorm on <math>X,</math> then the closure of <math>\{x \in X : p(x) < r\}</math> in <math>X</math> is equal to <math>\{x \in X : p(x) \leq r\}.</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li>
<li>The closure of <math>\{0\}</math> in a locally convex space <math>X</math> whose topology is defined by a family of continuous seminorms <math>\mathcal{P}</math> is equal to <math>\bigcap_{p \in \mathcal{P}} p^{-1}(0).</math>{{sfn|Narici|Beckenstein|2011|pp=149-153}}</li>
<li>A subset <math>S</math> in a seminormed space <math>(X, p)</math> is [[Bounded set (topological vector space)|bounded]] if and only if <math>p(S)</math> is bounded.{{sfn|Wilansky|2013|pp=49-50}}</li>
<li>If <math>(X, p)</math> is a seminormed space then the locally convex topology that <math>p</math> induces on <math>X</math> makes <math>X</math> into a [[Metrizable topological vector space|pseudometrizable TVS]] with a canonical pseudometric given by <math>d(x, y) := p(x - y)</math> for all <math>x, y \in X.</math>{{sfn|Narici|Beckenstein|2011|pp=115-154}}</li>
<li>The product of infinitely many seminormable spaces is again seminormable if and only if all but finitely many of these spaces are trivial (that is, 0-dimensional).{{sfn|Narici|Beckenstein|2011|pp=156–175}}</li>
</ul>