Definitions

# 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

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

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

$p\left(x\right) = g\left(x/|x|\right) |x| ,$

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.

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