Equivalence of metrics

In the study of metric spaces in mathematics, there are various notions of two metrics on the same underlying space being "the same", or equivalent.

In the following, will denote a non-empty set and and will denote two metrics on .

Topological equivalence

The two metrics and are said to be topologically equivalent if they generate the same topology on . The adjective "topological" is often dropped.[1] There are multiple ways of expressing this condition:

and

The following are sufficient but not necessary conditions for topological equivalence:

Strong equivalence

Two metrics and are strongly equivalent if and only if there exist positive constants and such that, for every ,

In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in , rather than potentially different constants associated with each point of .

Strong equivalence of two metrics implies topological equivalence, but not vice versa. An intuitive reason why topological equivalence does not imply strong equivalence is that bounded sets under one metric are also bounded under a strongly equivalent metric, but not necessarily under a topologically equivalent metric.

When the two metrics are those induced by norms respectively, then strong equivalence is equivalent to the condition that, for all ,

In finite dimensional spaces, all metrics induced by the p-norm, including the euclidean metric, the taxicab metric, and the Chebyshev distance, are strongly equivalent.[3]

Even if two metrics are strongly equivalent, not all properties of the respective metric spaces are preserved. For instance, a function from the space to itself might be a contraction mapping under one metric, but not necessarily under a strongly equivalent one.[4]

Properties preserved by equivalence

Notes

  1. Bishop and Goldberg, p. 10.
  2. Ok, p. 127, footnote 12.
  3. Ok, p. 138.
  4. Ok, p. 175.
  5. Ok, p. 209.
  6. Cartan, p. 27.

References

  • Richard L. Bishop, Samuel I. Goldberg (1980). Tensor analysis on manifolds. Dover Publications. 
  • Efe Ok (2007). Real analysis with economics applications. Princeton University Press. ISBN 0-691-11768-3. 
  • Henri Cartan (1971). Differential Calculus. Kershaw Publishing Company LTD. ISBN 0-395-12033-0. 
This article is issued from Wikipedia - version of the 10/22/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.