In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.
Definition
editAn 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle p} 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
editOn 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
editIf 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.