Asymmetric norm
In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.
Definition
An asymmetric norm on a real vector space [math]\displaystyle{ X }[/math] is a function [math]\displaystyle{ p : X \to [0, +\infty) }[/math] that has the following properties:
- Subadditivity, or the triangle inequality: [math]\displaystyle{ p(x + y) \leq p(x) + p(y) \text{ for all } x, y \in X. }[/math]
- Nonnegative homogeneity: [math]\displaystyle{ p(rx) = r p(x) \text{ for all } x \in X }[/math] and every non-negative real number [math]\displaystyle{ r \geq 0. }[/math]
- Positive definiteness: [math]\displaystyle{ p(x) \gt 0 \text{ unless } x = 0 }[/math]
Asymmetric norms differ from norms in that they need not satisfy the equality [math]\displaystyle{ p(-x) = p(x). }[/math]
If the condition of positive definiteness is omitted, then [math]\displaystyle{ p }[/math] is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for [math]\displaystyle{ x \neq 0, }[/math] at least one of the two numbers [math]\displaystyle{ p(x) }[/math] and [math]\displaystyle{ p(-x) }[/math] is not zero.
Examples
On the real line [math]\displaystyle{ \R, }[/math] the function [math]\displaystyle{ p }[/math] given by [math]\displaystyle{ 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]\displaystyle{ X, }[/math] the Minkowski functional [math]\displaystyle{ p_B }[/math] of a convex subset [math]\displaystyle{ B\subseteq X }[/math] that contains the origin is defined by the formula [math]\displaystyle{ p_B(x) = \inf \left\{r \geq 0: x \in r B \right\}\, }[/math] for [math]\displaystyle{ x \in X }[/math]. This functional is an asymmetric seminorm if [math]\displaystyle{ B }[/math] is an absorbing set, which means that [math]\displaystyle{ \bigcup_{r \geq 0} r B = X, }[/math] and ensures that [math]\displaystyle{ p(x) }[/math] is finite for each [math]\displaystyle{ x \in X. }[/math]
Corresponce between asymmetric seminorms and convex subsets of the dual space
If [math]\displaystyle{ B^* \subseteq \R^n }[/math] is a convex set that contains the origin, then an asymmetric seminorm [math]\displaystyle{ p }[/math] can be defined on [math]\displaystyle{ \R^n }[/math] by the formula [math]\displaystyle{ p(x) = \max_{\varphi \in B^*} \langle\varphi, x \rangle. }[/math] For instance, if [math]\displaystyle{ B^* \subseteq \R^2 }[/math] is the square with vertices [math]\displaystyle{ (\pm 1,\pm 1), }[/math] then [math]\displaystyle{ p }[/math] is the taxicab norm [math]\displaystyle{ 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]\displaystyle{ \R^n }[/math] can be obtained from some convex set, called its dual unit ball. Therefore, asymmetric seminorms are in one-to-one correspondence with convex sets that contain the origin. The seminorm [math]\displaystyle{ p }[/math] is
- positive definite if and only if [math]\displaystyle{ B^* }[/math] contains the origin in its topological interior,
- degenerate if and only if [math]\displaystyle{ B^* }[/math] is contained in a linear subspace of dimension less than [math]\displaystyle{ n, }[/math] and
- symmetric if and only if [math]\displaystyle{ B^* = -B^*. }[/math]
More generally, if [math]\displaystyle{ X }[/math] is a finite-dimensional real vector space and [math]\displaystyle{ B^* \subseteq X^* }[/math] is a compact convex subset of the dual space [math]\displaystyle{ X^* }[/math] that contains the origin, then [math]\displaystyle{ p(x) = \max_{\varphi \in B^*} \varphi(x) }[/math] is an asymmetric seminorm on [math]\displaystyle{ X. }[/math]
See also
- Finsler manifold
- Minkowski functional – Function made from a set
References
- Cobzaş, S. (2006). "Compact operators on spaces with asymmetric norm". Stud. Univ. Babeş-Bolyai Math. 51 (4): 69–87. ISSN 0252-1938. Bibcode: 2006math......8031C.
- S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Basel: Birkhäuser, 2013; ISBN:978-3-0348-0477-6.
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