Editing Seminorm (section)

===Pseudometrics and the induced topology===

A seminorm <math>p</math> on <math>X</math> induces a topology, called the {{em|seminorm-induced topology}}, via the canonical [[translation-invariant]] [[Pseudometric space|pseudometric]] <math>d_p : X \times X \to \R</math>; <math>d_p(x, y) := p(x - y) = p(y - x).</math> 
This topology is [[Hausdorff space|Hausdorff]] if and only if <math>d_p</math> is a metric, which occurs if and only if <math>p</math> is a [[Norm (mathematics)|norm]].{{sfn|Wilansky|2013 |pp=15-21}} 
This topology makes <math>X</math> into a [[Locally convex topological vector space|locally convex]] [[Metrizable topological vector space|pseudometrizable]] [[topological vector space]] that has a [[Bounded set (topological vector space)|bounded]] neighborhood of the origin and a [[neighborhood basis]] at the origin consisting of the following open balls (or the closed balls) centered at the origin:
<math display=block>\{x \in X : p(x) < r\} \quad \text{ or } \quad \{x \in X : p(x) \leq r\}</math>
as <math>r > 0</math> ranges over the positive reals. 
Every seminormed space <math>(X, p)</math> should be assumed to be endowed with this topology unless indicated otherwise. A topological vector space whose topology is induced by some seminorm is called {{em|seminormable}}. 

Equivalently, every vector space <math>X</math> with seminorm <math>p</math> induces a [[Quotient space (linear algebra)|vector space quotient]] <math>X / W,</math> where <math>W</math> is the subspace of <math>X</math> consisting of all vectors <math>x \in X</math> with <math>p(x) = 0.</math> Then <math>X / W</math> carries a norm defined by <math>p(x + W) = p(x).</math>  The resulting topology, [[Pullback|pulled back]] to <math>X,</math> is precisely the topology induced by <math>p.</math>

Any seminorm-induced topology makes <math>X</math> [[Locally convex topological vector space|locally convex]], as follows.  If <math>p</math> is a seminorm on <math>X</math> and <math>r \in \R,</math> call the set <math>\{x \in X : p(x) < r\}</math> the {{em|open ball of radius <math>r</math> about the origin}}; likewise the closed ball of radius <math>r</math> is <math>\{x \in X : p(x) \leq r\}.</math> The set of all open (resp. closed) <math>p</math>-balls at the origin forms a neighborhood basis of [[Convex set|convex]] [[Balanced set|balanced]] sets that are open (resp. closed) in the <math>p</math>-topology on <math>X.</math>

====Stronger, weaker, and equivalent seminorms====

The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker [[Norm (mathematics)|norms]].  If <math>p</math> and <math>q</math> are seminorms on <math>X,</math> then we say that <math>q</math> is {{em|stronger}} than <math>p</math> and that <math>p</math> is {{em|weaker}} than <math>q</math> if any of the following equivalent conditions holds:

# The topology on <math>X</math> induced by <math>q</math> is finer than the topology induced by <math>p.</math>
# If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> is a sequence in <math>X,</math> then <math>q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0</math> in <math>\R</math> implies <math>p\left(x_{\bull}\right) \to 0</math> in <math>\R.</math>{{sfn|Wilansky|2013 |pp=15-21}}
# If <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is a [[Net (mathematics)|net]] in <math>X,</math> then <math>q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i \in I} \to 0</math> in <math>\R</math> implies <math>p\left(x_{\bull}\right) \to 0</math> in <math>\R.</math>
# <math>p</math> is bounded on <math>\{x \in X : q(x) < 1\}.</math>{{sfn|Wilansky|2013 |pp=15-21}}
# If <math>\inf{} \{q(x) : p(x) = 1, x \in X\} = 0</math> then <math>p(x) = 0</math> for all <math>x \in X.</math>{{sfn|Wilansky|2013 |pp=15-21}}
# There exists a real <math>K > 0</math> such that <math>p \leq K q</math> on <math>X.</math>{{sfn|Wilansky|2013 |pp=15-21}}

The seminorms <math>p</math> and <math>q</math> are called {{em|equivalent}} if they are both weaker (or both stronger) than each other.  This happens if they satisfy any of the following conditions:
<ol>
<li>The topology on <math>X</math> induced by <math>q</math> is the same as the topology induced by <math>p.</math></li>
<li><math>q</math> is stronger than <math>p</math> and <math>p</math> is stronger than <math>q.</math>{{sfn|Wilansky|2013|pp=15-21}}</li>
<li>If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> is a sequence in <math>X</math> then <math>q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0</math> if and only if <math>p\left(x_{\bull}\right) \to 0.</math></li>
<li>There exist positive real numbers <math>r > 0</math> and <math>R > 0</math> such that <math>r q \leq p \leq R q.</math></li>
</ol>