equivalence of metrics

The two metrics $d\_1$ and $d\_2$ are said to be topologically equivalent if they generate the same topology on $X$. The adverb topologically is often dropped.Bishop and Goldberg, p. 10. There are multiple ways of expressing this condition:

• a subset $A\; \backslash subseteq\; X$ is $d\_1$-open if and only if it is $d\_2$-open;
• the open balls "nest": for any point $x\; \backslash in\; X$ and any radius $r\; >\; 0$, there exist radii $r\text{'},\; r$ > 0 such that $$B\_\{r\text{'}\}\; (x;\; d\_1)\; \backslash subseteq\; B\_r\; (x;\; d\_2)\; \backslash text\{\; and\; \}\; B\_\{r$$} (x; d_2) \subseteq B_r (x; d_1).
• the identity function $I\; :\; (X,d\_1)\; \backslash to\; (X,d\_2)$ is continuous with continuous inverse; that is, it is a homeomorphism.

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

• there exists a strictly increasing, continuous, and subadditive $f:\; \backslash R\; \backslash to\; \backslash R\_+$ such that $d\_2\; =\; f\; \backslash circ\; d\_1$.Ok, p. 137, footnote 12.
• for each $x\; \backslash in\; X$, there exist positive constants $\backslash alpha$ and $\backslash beta$ such that, for every point $y\; \backslash in\; X$, $$\backslash alpha\; d\_1\; (x,\; y)\; \backslash leq\; d\_2\; (x,\; y)\; \backslash leq\; \backslash beta\; d\_1\; (x,\; y).$$

Two metrics $d\_1$ and $d\_2$ on {{mvar|X}} are strongly or bilipschitz equivalent or uniformly equivalent if and only if there exist positive constants $\backslash alpha$ and $\backslash beta$ such that, for every $x,y\backslash in\; X$,

:$\backslash alpha\; d\_1(x,y)\; \backslash leq\; d\_2(x,y)\; \backslash leq\; \backslash beta\; d\_1\; (x,\; y).$

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 $X$, rather than potentially different constants associated with each point of $X$.

Strong equivalence of two metrics implies topological equivalence, but not vice versa. For example, the metrics $d\_1(x,y)=|x-y|$ and $d\_2(x,y)=|\backslash tan(x)-\backslash tan(y)|$ on the interval $\backslash left(-\backslash frac\{\backslash pi\}\{2\},\backslash frac\{\backslash pi\}\{2\}\backslash right)$ are topologically equivalent, but not strongly equivalent. In fact, this interval is bounded under one of these metrics but not the other. On the other hand, strong equivalences always take bounded sets to bounded sets.

When {{mvar|X}} is a vector space and the two metrics $d\_1$ and $d\_2$ are those induced by norms $\backslash |\backslash cdot\; \backslash |\_A$ and $\backslash |\backslash cdot\backslash |\_B$, respectively, then strong equivalence is equivalent to the condition that, for all $x\; \backslash in\; X$,

$$\backslash alpha\backslash |x\backslash |\_A\; \backslash leq\; \backslash |x\backslash |\_B\; \backslash leq\; \backslash beta\backslash |x\backslash |\_A$$

For linear operators between normed vector spaces, Lipschitz continuity is equivalent to continuity—an operator satisfying either of these conditions is called bounded.{{sfn|Carothers|2000|loc=Theorem 8.20}} Therefore, in this case, $d\_1$ and $d\_2$ are topologically equivalent if and only if they are strongly equivalent; the norms $\backslash |\backslash cdot\; \backslash |\_A$ and $\backslash |\backslash cdot\backslash |\_B$ are simply said to be equivalent.

In finite dimensional vector spaces, all metrics induced by a norm, including the euclidean metric, the taxicab metric, and the Chebyshev distance, are equivalent.{{sfn|Carothers|2000|loc=Theorem 8.22}}

• The continuity of a function is preserved if either the domain or range is remetrized by an equivalent metric, but uniform continuity is preserved only by strongly equivalent metrics.Ok, p. 209.
• The differentiability of a function $f:U\backslash to\; V$, for $V$ a normed space and $U$ a subset of a normed space, is preserved if either the domain or range is renormed by a strongly equivalent norm.Cartan, p. 27.
• A metric that is strongly equivalent to a complete metric is also complete; the same is not true of equivalent metrics because homeomorphisms do not preserve completeness. For example, since $(0,1)$ and $\backslash mathbb\; R$ are homeomorphic, the homeomorphism induces a metric on $(0,1)$ which is complete because $\backslash mathbb\; R$ is, and generates the same topology as the usual one, yet $(0,1)$ with the usual metric is not complete, because the sequence $(2^\{-n\})\_\{n\backslash in\backslash mathbb\; N\}$ is Cauchy but not convergent. (It is not Cauchy in the induced metric.)

{{reflist}}

{{refbegin}}

• {{cite book

| author = Richard L. Bishop

| author-link = Richard L. Bishop

| author2 = Samuel I. Goldberg

| title = Tensor analysis on manifolds

| year = 1980

| publisher = Dover Publications

| url = https://archive.org/details/tensoranalysison00bish

| url-access = registration

}}

• {{cite book |last1=Carothers |first1=N. L. |title=Real analysis |date=2000 |publisher=Cambridge University Press |isbn=0-521-49756-6}}
• {{cite book

| author = Henri Cartan

| author-link = Henri Cartan

| title = Differential Calculus

| year = 1971

| publisher = Kershaw Publishing Company LTD.

| isbn = 0-395-12033-0

}}

• {{cite book

| author = Efe Ok

| title = Real analysis with economics applications

| year = 2007

| publisher = Princeton University Press

| isbn = 0-691-11768-3

}}

{{refend}}

{{Metric spaces}}

Category:Metric geometry

Category:Equivalence (mathematics)