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
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