Asymmetric norm

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In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.

Definition[edit]

Let X be a real vector space. Then an asymmetric norm on X is a function p : X → R satisfying the following properties:

Examples[edit]

  • On the real line R, the function p given by
is an asymmetric norm but not a norm.
  • More generally, given a convex absorbing subset K of a real vector space containing no non-zero subspace, the Minkowski functional p given by
is an asymmetric norm but not necessarily a norm, unless K is also balanced.

References[edit]

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