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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

An asymmetric norm on a real vector space V is a function that has the following properties:

Asymmetric norms differ from norms in that they need not satisfy the equality p(-v) = p(v).

If the condition of positive definiteness is omitted, then p is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for v ≠ 0, at least one of the two numbers p(v) and p(-v) is not zero.

Examples

is an asymmetric norm but not a norm.
  • In a real vector space , the Minkowski functional of a convex subset that contains the origin is defined by the formula
for
This functional is an asymmetric seminorm if is an absorbing set, which means that , and ensures that is finite for each .

Corresponce between asymmetric seminorms and convex subsets of the dual space

If is a convex set that contains the origin, then an asymmetric seminorm can be defined on by the formula

.

For instance, if is the square with vertices , then is the taxicab norm . Different convex sets yield different seminorms, and every asymmetric seminorm on can be obtained from some convex set, called its dual unit ball. Therefore, asymmetric seminorms are in one-to-one correspondence with convex sets that contain the origin. The seminorm is

  • positive definite if and only if contains the origin in its 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 .

References

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