Editing Seminorm (section)

==Algebraic properties==

Every seminorm is a [[sublinear function]], and thus satisfies all [[Sublinear_function#Properties|properties of a sublinear function]], including [[convex function|convexity]], <math>p(0) = 0,</math> and for all vectors <math>x, y \in X</math>:
the [[reverse triangle inequality]]: {{sfn|Narici|Beckenstein|2011|pp=120-121}}{{sfn|Narici|Beckenstein|2011|pp=177-220}}
<math display=block>|p(x) - p(y)| \leq p(x - y)</math>
and also
<math display=inline>0 \leq \max \{p(x), p(-x)\}</math> and <math>p(x) - p(y) \leq p(x - y).</math>{{sfn|Narici|Beckenstein|2011|pp=120-121}}{{sfn|Narici|Beckenstein|2011|pp=177-220}}

For any vector <math>x \in X</math> and positive real <math>r > 0:</math>{{sfn|Narici|Beckenstein|2011|pp=116−128}}
<math display=block>x + \{y \in X : p(y) < r\} = \{y \in X : p(x - y) < r\}</math>
and furthermore, <math>\{x \in X : p(x) < r\}</math> is an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}}

If <math>p</math> is a sublinear function on a real vector space <math>X</math> then there exists a linear functional <math>f</math> on <math>X</math> such that <math>f \leq p</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}} and furthermore, for any linear functional <math>g</math> on <math>X,</math> <math>g \leq p</math> on <math>X</math> if and only if <math>g^{-1}(1) \cap \{x \in X : p(x) < 1\} = \varnothing.</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}

'''Other properties of seminorms'''

Every seminorm is a [[balanced function]]. 
A seminorm <math>p</math> is a norm on <math>X</math> if and only if <math>\{x \in X : p(x) < 1\}</math> does not contain a non-trivial vector subspace. 

If <math>p : X \to [0, \infty)</math> is a seminorm on <math>X</math> then <math>\ker p := p^{-1}(0)</math> is a vector subspace of <math>X</math> and for every <math>x \in X,</math> <math>p</math> is constant on the set <math>x + \ker p = \{x + k : p(k) = 0\}</math> and equal to <math>p(x).</math><ref group=proof name=ConstantOnEquivClasses>Let <math>x \in X</math> and <math>k \in p^{-1}(0).</math> It remains to show that <math>p(x + k) = p(x).</math> The triangle inequality implies <math>p(x + k) \leq p(x) + p(k) = p(x) + 0 = p(x).</math> Since <math>p(-k) = 0,</math> <math>p(x) = p(x) - p(-k) \leq p(x - (-k)) = p(x + k),</math> as desired. <math>\blacksquare</math></ref> 

Furthermore, for any real <math>r > 0,</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}} 
<math display="block">r \{x \in X : p(x) < 1\} = \{x \in X : p(x) < r\} = \left\{x \in X : \tfrac{1}{r} p(x) < 1 \right\}.</math>

If <math>D</math> is a set satisfying <math>\{x \in X : p(x) < 1\} \subseteq D \subseteq \{x \in X : p(x) \leq 1\}</math> then <math>D</math> is [[Absorbing set|absorbing]] in <math>X</math> and <math>p = p_D</math> where <math>p_D</math> denotes the [[Minkowski functional]] associated with <math>D</math> (that is, the gauge of <math>D</math>).{{sfn|Schaefer|Wolff|1999|p=40}} In particular, if <math>D</math> is as above and <math>q</math> is any seminorm on <math>X,</math> then <math>q = p</math> if and only if <math>\{x \in X : q(x) < 1\} \subseteq D \subseteq \{x \in X : q(x) \leq\}.</math>{{sfn|Schaefer|Wolff|1999|p=40}}

If <math>(X, \|\,\cdot\,\|)</math> is a normed space and <math>x, y \in X</math> then <math>\|x - y\| = \|x - z\| + \|z - y\|</math> for all <math>z</math> in the interval <math>[x, y].</math>{{sfn|Narici|Beckenstein|2011|pp=107-113}} 

Every norm is a [[convex function]] and consequently, finding a global maximum of a norm-based [[objective function]] is sometimes tractable.

===Relationship to other norm-like concepts===

Let <math>p : X \to \R</math> be a non-negative function. The following are equivalent:
<ol>
<li><math>p</math> is a seminorm.</li>
<li><math>p</math> is a [[Convex function|convex]] [[F-seminorm|<math>F</math>-seminorm]].</li>
<li><math>p</math> is a convex balanced [[Metrizable topological vector space|''G''-seminorm]].{{sfn|Schechter|1996|p=691}}</li>
</ol>

If any of the above conditions hold, then the following are equivalent:
<ol>
<li><math>p</math> is a norm;</li>
<li><math>\{x \in X : p(x) < 1\}</math> does not contain a non-trivial vector subspace.{{sfn|Narici|Beckenstein|2011|p=149}}</li>
<li>There exists a [[Normed vector space|norm]] on <math>X,</math> with respect to which, <math>\{x \in X : p(x) < 1\}</math> is bounded.</li>
</ol>

If <math>p</math> is a sublinear function on a real vector space <math>X</math> then the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=177-220}} 
<ol>
<li><math>p</math> is a [[linear functional]];</li>
<li><math>p(x) + p(-x) \leq 0 \text{ for every } x \in X</math>;</li>
<li><math>p(x) + p(-x) = 0 \text{ for every } x \in X</math>;</li>
</ol>

===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>

===Hahn–Banach theorem for seminorms===

Seminorms offer a particularly clean formulation of the [[Hahn–Banach theorem]]: 
:If <math>M</math> is a vector subspace of a seminormed space <math>(X, p)</math> and if <math>f</math> is a continuous linear functional on <math>M,</math> then <math>f</math> may be extended to a continuous linear functional <math>F</math> on <math>X</math> that has the same norm as <math>f.</math>{{sfn|Wilansky|2013|pp=21-26}}

A similar extension property also holds for seminorms:

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=150}}{{sfn|Wilansky|2013|pp=18-21}}|note=Extending seminorms|math_statement=
If <math>M</math> is a vector subspace of <math>X,</math> <math>p</math> is a seminorm on <math>M,</math> and <math>q</math> is a seminorm on <math>X</math> such that <math>p \leq q\big\vert_M,</math> then there exists a seminorm <math>P</math> on <math>X</math> such that <math>P\big\vert_M = p</math> and <math>P \leq q.</math> 
}}

:'''Proof''': Let <math>S</math> be the [[convex hull]] of <math>\{m \in M : p(m) \leq 1\} \cup \{x \in X : q(x) \leq 1\}.</math> Then <math>S</math> is an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in <math>X</math> and so the [[Minkowski functional]] <math>P</math> of <math>S</math> is a seminorm on <math>X.</math> This seminorm satisfies <math>p = P</math> on <math>M</math> and <math>P \leq q</math> on <math>X.</math> <math>\blacksquare</math>