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

::

:is an asymmetric norm but not a norm.

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

::$p(x)\; =\; g(x/|x|)\; |x|\; ,$

:is an asymmetric norm on R^"n" but not necessarily a norm.

References

* cite journal
last = Cobzaş
first = S.
title = Compact operators on spaces with asymmetric norm
journal = Stud. Univ. Babeş-Bolyai Math.
volume= 51
year = 2006
issue = 4
pages = 69–87
issn = 0252-1938 MathSciNet|id=2314639