Asymmetric norm

An asymmetric norm on a real vector space V is a function ${\displaystyle p:V\to [0,+\infty )}$ 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.

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

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

is an asymmetric norm but not a norm.

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

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

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

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

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

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

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

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

• 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.