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 V is a function $p:V\to [0,+\infty )$ that has the following properties:

• Subadditivity, or the triangle inequality: p(v + w) ≤ p(v) + p(w) for every two vectors v,w ∈ V.
• Homogeneity: p(λv) = λp(v) for every vector v ∈ X and every non-negative real number λ ≥ 0.
• Positive definiteness: p(v) > 0 unless v = 0.

Asymmetric norms differ from norms in that they need not satisfy the equality p(-v) = p(v).

If the condition of positive definiteness is omitted, then p is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for v ≠ 0, at least one of the two numbers p(v) and p(-v) is not zero.

Examples

• On the real line R, the function p given by

$p(x)={\begin{cases}|x|,&x\leq 0;\\2|x|,&x\geq 0;\end{cases}}$

is an asymmetric norm but not a norm.

• In a real vector space $V$, the Minkowski functional $p_{B}$ of a convex subset $B\subseteq V$ that contains the origin is defined by the formula

$p_{B}(v)=\inf \left\{r\geq 0:v\in rB\right\}\,$ for $v\in V$

This functional is an asymmetric seminorm if $B$ is an absorbing set, which means that $\textstyle {\bigcup _{r\geq 0}rB=V}$, and ensures that $p(v)$ is finite for each $v\in V$.

Corresponce between asymmetric seminorms and convex subsets of the dual space

If $B^{*}\subseteq \mathbb {R} ^{n}$ is a convex set that contains the origin, then an asymmetric seminorm $p$ can be defined on $\mathbb {R} ^{n}$ by the formula

$\textstyle {p(v)=\max _{\varphi \in B^{*}}\langle \varphi ,v\rangle }$.

For instance, if $B^{*}\subseteq \mathbb {R} ^{2}$ is the square with vertices $(\pm 1,\pm 1)$, then $p$ is the taxicab norm $v=(v_{0},v_{1})\mapsto |v_{0}|+|v_{1}|$. Different convex sets yield different seminorms, and every asymmetric seminorm on $\mathbb {R} ^{n}$ 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 $p$ is

• positive definite if and only if $B^{*}$ contains the origin in its interior,
• degenerate if and only if $B^{*}$ is contained in a linear subspace of dimension less than $n$, and
• symmetric if and only if $B^{*}=-B^{*}$.

More generally, if $V$ is a finite-dimensional real vector space and $B^{*}\subseteq V^{*}$ is a compact convex subset of the dual space $V^{*}$ that contains the origin, then $\textstyle {p(v)=\max _{\varphi \in B^{*}}\varphi (v)}$ is an asymmetric seminorm on $V$.

References

• Cobzaş, S. (2006). "Compact operators on spaces with asymmetric norm". Stud. Univ. Babeş-Bolyai Math. 51 (4): 69–87. 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.