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In [[mathematics]], an '''asymmetric norm''' on a [[vector space]] is a generalization of the concept of a [[norm (mathematics)|norm]].

==Definition==

Let ''X'' be a [[real number|real]] vector space. Then an '''asymmetric norm''' on ''X'' is a [[function (mathematics)|function]] ''p'' : ''X'' → '''R''' satisfying the following properties:

* non-negativity: for all ''x'' ∈ ''X'', ''p''(''x'') ≥ 0;
* [[Positive-definite function|definiteness]]: for ''x'' ∈ ''X'', ''x'' = 0 [[if and only if]] ''p''(''x'') = ''p''(−''x'') = 0;
* [[homogeneous function|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

::<math>p(x) = \begin{cases} |x|, & x \leq 0; \\ 2 |x|, & x \geq 0; \end{cases}</math>

: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

::<math>p(x) = g(x/|x|) |x| \,</math>

: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
| mr = 2314639
}}

[[Category:Linear algebra]]
[[Category:Norms (mathematics)]]

{{Linear-algebra-stub}}

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