View source for Asymmetric norm

{{short description|Generalization of the concept of a norm}}
In [[mathematics]], an '''asymmetric norm''' on a [[vector space]] is a generalization of the concept of a [[Norm (mathematics)|norm]].

==Definition==

An '''asymmetric norm''' on a [[Real number|real]] [[vector space]] <math>X</math> is a [[Function (mathematics)|function]] <math>p : X \to [0, +\infty)</math> that has the following properties:

* '''[[Subadditivity]]''', or the [[triangle inequality]]: <math>p(x + y) \leq p(x) + p(y) \text{ for all } x, y \in X.</math>
* '''[[Nonnegative homogeneity]]''': <math>p(rx) = r p(x) \text{ for all } x \in X</math> and every non-negative real number <math>r \geq 0.</math>
* '''[[Positive-definite function|Positive definiteness]]''': <math>p(x) > 0 \text{ unless } x = 0</math>

Asymmetric norms differ from [[Norm (mathematics)|norms]] in that they need not satisfy the equality <math>p(-x) = p(x).</math> 

If the condition of positive definiteness is omitted, then <math>p</math> is an '''asymmetric seminorm'''. A weaker condition than positive definiteness is '''non-degeneracy''': that for <math>x \neq 0,</math> at least one of the two numbers <math>p(x)</math> and <math>p(-x)</math> is not zero.

== Examples ==

On the [[real line]] <math>\R,</math> the function <math>p</math> given by
<math display="block">p(x) = \begin{cases}|x|, & x \leq 0; \\ 2 |x|, & x \geq 0; \end{cases}</math>
is an asymmetric norm but not a norm.

In a real vector space <math>X,</math> the {{em|[[Minkowski functional]]}} <math>p_B</math> of a convex subset <math>B\subseteq X</math> that contains the origin is defined by the formula
<math display="block">p_B(x) = \inf \left\{r \geq 0: x \in r B \right\}\,</math> for <math>x \in X</math>.
This functional is an asymmetric seminorm if <math>B</math> is an absorbing set, which means that <math>\bigcup_{r \geq 0} r B = X,</math> and ensures that <math>p(x)</math> is finite for each <math>x \in X.</math>

== Corresponce between asymmetric seminorms and convex subsets of the dual space ==

If <math>B^* \subseteq \R^n</math> is a [[convex set]] that contains the origin, then an asymmetric seminorm <math>p</math> can be defined on <math>\R^n</math> by the formula
<math display="block">p(x) = \max_{\varphi \in B^*} \langle\varphi, x \rangle.</math>
For instance, if <math>B^* \subseteq \R^2</math> is the square with vertices <math>(\pm 1,\pm 1),</math> then <math>p</math> is the [[taxicab norm]] <math>x = \left(x_0, x_1\right) \mapsto \left|x_0\right| + \left|x_1\right|.</math> Different convex sets yield different seminorms, and every asymmetric seminorm on <math>\R^n</math> can be obtained from some convex set, called its '''dual unit ball'''. Therefore, asymmetric seminorms are in [[Bijection|one-to-one correspondence]] with convex sets that contain the origin. The seminorm <math>p</math> is 
* positive definite if and only if <math>B^*</math> contains the origin in its [[Interior (mathematics)|topological interior]],
* degenerate if and only if <math>B^*</math> is contained in a [[linear subspace]] of dimension less than <math>n,</math> and
* symmetric if and only if <math>B^* = -B^*.</math>

More generally, if <math>X</math> is a [[finite-dimensional]] real vector space and <math>B^* \subseteq X^*</math> is a compact convex subset of the [[dual space]] <math>X^*</math> that contains the origin, then <math>p(x) = \max_{\varphi \in B^*} \varphi(x)</math> is an asymmetric seminorm on <math>X.</math>

== See also ==

* {{annotated link|Finsler manifold}}
* {{annotated link|Minkowski functional}}

==References==

* {{cite journal|last=Cobzaş|first=S.|title=Compact operators on spaces with asymmetric norm|journal=Stud. Univ. Babeş-Bolyai Math.|volume=51|year=2006|issue=4|pages=69&ndash;87|arxiv=math/0608031 |bibcode=2006math......8031C |issn=0252-1938|mr=2314639}}
* S. Cobzas, ''Functional Analysis in Asymmetric Normed Spaces'', Frontiers in Mathematics, Basel: Birkhäuser, 2013; {{ISBN|978-3-0348-0477-6}}.  

[[Category:Linear algebra]]
[[Category:Norms (mathematics)]]

{{Functional analysis}}
{{Topological vector spaces}}

{{Linear-algebra-stub}}