Asymmetric norm

In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.

Definition

Let X be a real vector space. Then an asymmetric norm on X is a function p : X → R satisfying the following properties:

non-negativity: for all x ∈ X, p(x) ≥ 0;
definiteness: for x ∈ X, x = 0 if and only if p(x) = p(−x) = 0;
homogeneity: for all x ∈ X and all λ ≥ 0, p(λx) = λp(x);
the triangle inequality: for all x, y ∈ X, p(x + y) ≤ p(x) + p(y).

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.

More generally, given a strictly positive function g : Sⁿ⁻¹ → R defined on the unit sphere Sⁿ⁻¹ in Rⁿ (with respect to the usual Euclidean norm |·|, say), the function p given by

p(x) = g(x/|x|) |x| \,

is an asymmetric norm on Rⁿ but not necessarily a norm.

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.