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- 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).
- 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 : 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.
- Cobzaş, S. (2006). "Compact operators on spaces with asymmetric norm". Stud. Univ. Babeş-Bolyai Math. 51 (4): 69–87. ISSN 0252-1938. MR 2314639.
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