Editing Seminorm (section)

==Definition==

Let <math>X</math> be a vector space over either the [[real number]]s <math>\R</math> or the [[Complex number|complex]] numbers <math>\Complex.</math> 
A [[real-valued function]] <math>p : X \to \R</math> is called a {{em|seminorm}} if it satisfies the following two conditions:

# [[Subadditive function|Subadditivity]]{{sfn|Kubrusly|2011|p=200}}/[[Triangle inequality]]: <math>p(x + y) \leq p(x) + p(y)</math> for all <math>x, y \in X.</math>
# [[Homogeneous function|Absolute homogeneity]]:{{sfn|Kubrusly|2011|p=200}} <math>p(s x) =|s|p(x)</math> for all <math>x \in X</math> and all scalars <math>s.</math>

These two conditions imply that <math>p(0) = 0</math><ref group="proof">If <math>z \in X</math> denotes the zero vector in <math>X</math> while <math>0</math> denote the zero scalar, then absolute homogeneity implies that <math>p(z) = p(0 z) = |0|p(z) = 0 p(z) = 0.</math> <math>\blacksquare</math></ref> and that every seminorm <math>p</math> also has the following property:<ref group="proof">Suppose <math>p : X \to \R</math> is a seminorm and let <math>x \in X.</math> Then absolute homogeneity implies <math>p(-x) = p((-1) x) =|-1|p(x) = p(x).</math> The triangle inequality now implies <math>p(0) = p(x + (- x)) \leq p(x) + p(-x) = p(x) + p(x) = 2 p(x).</math> Because <math>x</math> was an arbitrary vector in <math>X,</math> it follows that <math>p(0) \leq 2 p(0),</math> which implies that <math>0 \leq p(0)</math> (by subtracting <math>p(0)</math> from both sides). Thus <math>0 \leq p(0) \leq 2 p(x)</math> which implies <math>0 \leq p(x)</math> (by multiplying through by <math>1/2</math>). <math>\blacksquare</math></ref>
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<li>[[Nonnegative|Nonnegativity]]:{{sfn|Kubrusly|2011|p=200}} <math>p(x) \geq 0</math> for all <math>x \in X.</math></li>
</ol>

Some authors include non-negativity as part of the definition of "seminorm" (and also sometimes of "norm"), although this is not necessary since it follows from the other two properties. 

By definition, a [[Norm (mathematics)|norm]] on <math>X</math> is a seminorm that also separates points, meaning that it has the following additional property:
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<li>[[Positive definite]]/Positive{{sfn|Kubrusly|2011|p=200}}/{{visible anchor|Point-separating}}: whenever <math>x \in X</math> satisfies <math>p(x) = 0,</math> then <math>x = 0.</math></li>
</ol>

A {{em|{{visible anchor|seminormed space}}}} is a pair <math>(X, p)</math> consisting of a vector space <math>X</math> and a seminorm <math>p</math> on <math>X.</math> If the seminorm <math>p</math> is also a norm then the seminormed space <math>(X, p)</math> is called a {{em|[[normed space]]}}.

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a [[sublinear function]]. A map <math>p : X \to \R</math> is called a {{em|[[sublinear function]]}} if it is subadditive and [[positive homogeneous]]. Unlike a seminorm, a sublinear function is {{em|not}} necessarily nonnegative. Sublinear functions are often encountered in the context of the [[Hahn–Banach theorem]]. 
A real-valued function <math>p : X \to \R</math> is a seminorm if and only if it is a [[Sublinear function|sublinear]] and [[balanced function]].