Editing Seminorm (section)

===Inequalities involving seminorms===

If <math>p, q : X \to [0, \infty)</math> are seminorms on <math>X</math> then:
<ul>
<li><math>p \leq q</math> if and only if <math>q(x) \leq 1</math> implies <math>p(x) \leq 1.</math>{{sfn|Narici|Beckenstein|2011|pp=149–153}}</li>
<li>If <math>a > 0</math> and <math>b > 0</math> are such that <math>p(x) < a</math> implies <math>q(x) \leq b,</math> then <math>a q(x) \leq b p(x)</math> for all <math>x \in X.</math> {{sfn|Wilansky|2013|pp=18-21}}</li>
<li>Suppose <math>a</math> and <math>b</math> are positive real numbers and <math>q, p_1, \ldots, p_n</math> are seminorms on <math>X</math> such that for every <math>x \in X,</math> if <math>\max \{p_1(x), \ldots, p_n(x)\} < a</math> then <math>q(x) < b.</math> Then <math>a q \leq b \left(p_1 + \cdots + p_n\right).</math>{{sfn|Narici|Beckenstein|2011|p=149}}</li>
<li>If <math>X</math> is a vector space over the reals and <math>f</math> is a non-zero linear functional on <math>X,</math> then <math>f \leq p</math> if and only if <math>\varnothing = f^{-1}(1) \cap \{x \in X : p(x) < 1\}.</math>{{sfn|Narici|Beckenstein|2011|pp=149–153}}</li>
</ul>

If <math>p</math> is a seminorm on <math>X</math> and <math>f</math> is a linear functional on <math>X</math> then:
<ul>
<li><math>|f| \leq p</math> on <math>X</math> if and only if <math>\operatorname{Re} f \leq p</math> on <math>X</math> (see footnote for proof).<ref>Obvious if <math>X</math> is a real vector space. For the non-trivial direction, assume that <math>\operatorname{Re} f \leq p</math> on <math>X</math> and let <math>x \in X.</math> Let <math>r \geq 0</math> and <math>t</math> be real numbers such that <math>f(x) = r e^{i t}.</math> Then <math>|f(x)|= r = f\left(e^{-it} x\right) = \operatorname{Re}\left(f\left(e^{-it} x\right)\right) \leq p\left(e^{-it} x\right) = p(x).</math></ref>{{sfn|Wilansky|2013|p=20}}</li>
<li><math>f \leq p</math> on <math>X</math> if and only if <math>f^{-1}(1) \cap \{x \in X : p(x) < 1 = \varnothing\}.</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}{{sfn|Narici|Beckenstein|2011|pp=149–153}}</li>
<li>If <math>a > 0</math> and <math>b > 0</math> are such that <math>p(x) < a</math> implies <math>f(x) \neq b,</math> then <math>a |f(x)| \leq b p(x)</math> for all <math>x \in X.</math>{{sfn|Wilansky|2013|pp=18-21}}</li>
</ul>