Editing Seminorm (section)

==Examples==

<ul>
<li>The {{em|trivial seminorm}} on <math>X,</math> which refers to the constant <math>0</math> map on <math>X,</math> induces the [[indiscrete topology]] on <math>X.</math></li>
<li>Let <math>\mu</math> be a measure on a space <math>\Omega</math>. For an arbitrary constant <math>c \geq 1</math>, let <math>X</math> be the set of all functions <math>f: \Omega \rightarrow \mathbb{R}</math> for which
<math display="block">\lVert f \rVert_c := \left( \int_{\Omega}| f |^c \, d\mu \right)^{1/c}</math>
exists and is finite. It can be shown that <math>X</math> is a vector space, and the functional <math>\lVert \cdot \rVert_c</math> is a seminorm on <math>X</math>. However, it is not always a norm (e.g. if <math>\Omega = \mathbb{R}</math> and <math>\mu</math> is the [[Lebesgue measure]]) because <math>\lVert h \rVert_c = 0</math> does not always imply <math>h = 0</math>. To make <math>\lVert \cdot \rVert_c</math> a norm, quotient <math>X</math> by the closed subspace of functions <math>h</math> with <math>\lVert h \rVert_c = 0</math>. The [[Lp_space#Lp_spaces_and_Lebesgue_integrals|resulting space]], <math>L^c(\mu)</math>, has a norm induced by <math>\lVert \cdot \rVert_c</math>.</li>
<li>If <math>f</math> is any [[linear form]] on a vector space then its [[absolute value]] <math>|f|,</math> defined by <math>x \mapsto |f(x)|,</math> is a seminorm.</li>
<li>A [[sublinear function]] <math>f : X \to \R</math> on a real vector space <math>X</math> is a seminorm if and only if it is a {{em|symmetric function}}, meaning that <math>f(-x) = f(x)</math> for all <math>x \in X.</math></li>
<li>Every real-valued [[sublinear function]] <math>f : X \to \R</math> on a real vector space <math>X</math> induces a seminorm <math>p : X \to \R</math> defined by <math>p(x) := \max \{f(x), f(-x)\}.</math>{{sfn|Narici|Beckenstein|2011|pp=120–121}}</li>
<li>Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a [[vector subspace]] is once again a seminorm (respectively, norm).</li>
<li>If <math>p : X \to \R</math> and <math>q : Y \to \R</math> are seminorms (respectively, norms) on <math>X</math> and <math>Y</math> then the map <math>r : X \times Y \to \R</math> defined by <math>r(x, y) = p(x) + q(y)</math> is a seminorm (respectively, a norm) on <math>X \times Y.</math> In particular, the maps on <math>X \times Y</math> defined by <math>(x, y) \mapsto p(x)</math> and <math>(x, y) \mapsto q(y)</math> are both seminorms on <math>X \times Y.</math></li>
<li>If <math>p</math> and <math>q</math> are seminorms on <math>X</math> then so are{{sfn|Narici|Beckenstein|2011|pp=116–128}} 
<math display="block">(p \vee q)(x) = \max \{p(x), q(x)\}</math> and <math display="block">(p \wedge q)(x) := \inf \{p(y) + q(z) : x = y + z \text{ with } y, z \in X\}</math>
where <math>p \wedge q \leq p</math> and <math>p \wedge q \leq q.</math>{{sfn|Wilansky|2013|pp=15-21}}
</li>
<li>The space of seminorms on <math>X</math> is generally not a [[distributive lattice]] with respect to the above operations. For example, over <math>\R^2</math>, <math>p(x, y) := \max(|x|, |y|), q(x, y) := 2|x|, r(x, y) := 2|y| </math> are such that 
<math display="block">((p \vee q) \wedge (p \vee r)) (x, y) = \inf \{\max(2|x_1|, |y_1|) + \max(|x_2|, 2|y_2|) : x = x_1 + x_2 \text{ and } y = y_1 + y_2\}</math> while <math>(p \vee q \wedge r) (x, y) := \max(|x|, |y|)</math></li>
<li>If <math>L : X \to Y</math> is a [[linear map]] and <math>q : Y \to \R</math> is a seminorm on <math>Y,</math> then <math>q \circ L : X \to \R</math> is a seminorm on <math>X.</math> The seminorm <math>q \circ L</math> will be a norm on <math>X</math> if and only if <math>L</math> is injective and the restriction <math>q\big\vert_{L(X)}</math> is a norm on <math>L(X).</math></li>
</ul>