In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.
Definition edit
An asymmetric norm on a real vector space is a function that has the following properties:
- Subadditivity, or the triangle inequality:
- Nonnegative homogeneity: and every non-negative real number
- Positive definiteness:
Asymmetric norms differ from norms in that they need not satisfy the equality
If the condition of positive definiteness is omitted, then is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for at least one of the two numbers and is not zero.
Examples edit
On the real line the function given by
In a real vector space the Minkowski functional of a convex subset that contains the origin is defined by the formula
Corresponce between asymmetric seminorms and convex subsets of the dual space edit
If is a convex set that contains the origin, then an asymmetric seminorm can be defined on by the formula
- positive definite if and only if contains the origin in its topological interior,
- degenerate if and only if is contained in a linear subspace of dimension less than and
- symmetric if and only if
More generally, if is a finite-dimensional real vector space and is a compact convex subset of the dual space that contains the origin, then is an asymmetric seminorm on
See also edit
- Finsler manifold – smooth manifold equipped with a Minkowski functional at each tangent space
- Minkowski functional – Function made from a set
References edit
- Cobzaş, S. (2006). "Compact operators on spaces with asymmetric norm". Stud. Univ. Babeş-Bolyai Math. 51 (4): 69–87. arXiv:math/0608031. Bibcode:2006math......8031C. ISSN 0252-1938. MR 2314639.
- S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Basel: Birkhäuser, 2013; ISBN 978-3-0348-0477-6.