Strongly Equivalent Metrics¶

Definition¶

Let $X$ be a metric space. The metrics $d_p$ and $d_q$ on $X$ (two arbitrary metrics; the subscripts are only labels) are termed strongly equivalent, denoted $d_p\approx d_q$ $d_{q}$ , if positive constants $a$ and $b$ exist such that for any $x,y \in X$ $X$ ,

a \cdot d_p(x,y) \leq d_q(x,y) \leq b \cdot d_p(x,y).

Interpretation¶

Strongly equivalent metrics measure distances on different scales, but the ratio between the two distances is bounded above and below by fixed constants. A distance that is small in one metric is therefore small in the other, uniformly over all pairs of points.

Around a point x, the Euclidean ball, a diamond-shaped taxicab ball of the same radius inside it, and a smaller Euclidean ball inside the diamond, showing that each ball contains a ball of the other metric.

On $\mathbb{R}^2$ , the illustrated Euclidean and taxicab distances satisfy $d_2\le d_1\le\sqrt{2}\,d_2$ $2 d_{2}$ . In the figure, the diamond of points with $d_1$ -distance less than $r$ from $\boldsymbol{x}$ lies inside the disc of points with $d_2$ -distance less than $r$ , and it contains the smaller disc of $d_2$ -radius $r/\sqrt{2}$ . Being close to $\boldsymbol{x}$ in one metric therefore forces being close in the other, so the two metrics have the same convergent sequences.

The constants $a$ and $b$ must work for all $x,y$ at once. Because of this uniformity, strongly equivalent metrics share more than convergent sequences: they have the same Cauchy sequences, so one of them is complete if and only if the other is. For metrics that come from norms, strong equivalence is the same as equivalence of the norms, so any two $p$ -metrics on $\mathbb{R}^n$ are strongly equivalent.

Having the same convergent sequences is not enough for strong equivalence. On $(-\tfrac{\pi}{2},\tfrac{\pi}{2})$ , the metric $d_{\tan}(x,y)=|\tan x-\tan y|$ has the same convergent sequences as $|x-y|$ $y ∣$ , but the ratio $d_{\tan}(x,y)/|x-y|$ is unbounded near $\pm\tfrac{\pi}{2}$ , and $x_k=\tfrac{\pi}{2}-\tfrac{1}{k}$ is Cauchy for $|x-y|$ and not for $d_{\tan}$ , since $\tan x_k\to+\infty$ $+ \infty$ . Such metrics return later in the course as topologically equivalent metrics.

Key Results¶

The strong equivalence relation possesses symmetry ( $d_{p} \approx$ $d_p\approx d_q$ if and only if $d_q\approx d_p$ $d_{p}$ ), reflexivity ( $d_{p} \approx$ $d_p\approx d_p$ $d_{p}$ ) and transitivity (if $d_p\approx d_q$ and $d_q\approx d_r$ $d_{r}$ , then $d_p\approx d_r$ $d_{r}$ ).
Strongly equivalent metrics induce the same notion of convergence.

Examples¶

Any two $p$ -metrics on $\mathbb{R}^n$ are strongly equivalent.
The discrete metric, $d_{\mathrm{disc}}$ , is not strongly equivalent to any of the $p$ -metrics.

Prerequisites¶

Metric and Metric Space

Sources¶

Math III: Analysis · Lecture 4

Aliases: Lipschitz equivalent metrics, strongly equivalent Symbols: element of, infinity, pi, real numbers, square root

Last updated: 20 October 2026 03:02