Editing Seminorm (section)

====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>