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:
Examples
-
- 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
-
- 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. MR2314639.