Question

Show that the metrics $\rho(x,y) = \sqrt{d(x,y)}$ and $\tau(x,y) = \min\{d(x,y),1\}$ are equivalent to $d(x,y)$.

score 140 · Answer 1 · 2026-02-23 04:43:02

140

Views

Show that the metrics $\rho(x,y) = \sqrt{d(x,y)}$ and $\tau(x,y) = \min\{d(x,y),1\}$ are equivalent to $d(x,y)$.

Published on 23 Feb 2026 - 4:43

score 631 · Answer 2 · 2026-02-23 04:38:33

631

Views

Every two norms on a finite dimensional (real or complex) vector space $V$ are equivalent.

Published on 23 Feb 2026 - 4:38

score 769 · Answer 3 · 2026-02-23 04:39:35

769

Views

Are d1 and lift metric equivalent distances?

Published on 23 Feb 2026 - 4:39

score 837 · Answer 4 · 2026-02-23 04:41:39

837

Views

Show that $d$ is equivalent to the usual metric on $\mathbb{N}$ but that $(\mathbb{N},d)$ is not complete.

Published on 23 Feb 2026 - 4:41

score 515 · Answer 5 · 2026-02-23 04:38:02

515

Views

Prove that two metrics generate the same topology.

Published on 23 Feb 2026 - 4:38

score 543 · Answer 6 · 2026-02-23 04:43:04

543

Views

Equivalence of Euclidean metric and the metric $d(x,y) = min{1, |x-y|}$ on $\mathbb R^{2}$

Published on 23 Feb 2026 - 4:43

score 617 · Answer 7 · 2026-02-23 04:38:30

617

Views

Topology induced by equivalent norms

Published on 23 Feb 2026 - 4:38

score 3.4k · Answer 8 · 2026-02-23 04:41:37

3.4k

Views

Show that, for every metric $d$, the metrics $d/(1+d)$ and $\min\{1,d\}$ are equivalent

Published on 23 Feb 2026 - 4:41

score 48 · Answer 9 · 2026-02-23 04:34:42

48

Views

Metric to compare conditional probabilities

Published on 23 Feb 2026 - 4:34

score 62 · Answer 10 · 2026-02-23 04:42:59

62

Views

Existence of a metric on $\Bbb Q$ which is complete and equivalent to the usual metric.

Published on 23 Feb 2026 - 4:42

score 429 · Answer 11 · 2026-02-23 04:36:29

429

Views

Determine if two norms are equivalent.

Published on 23 Feb 2026 - 4:36

score 163 · Answer 12 · 2026-02-23 04:41:55

163

Views

How to prove criterion for topological equivalence of metrics?

Published on 23 Feb 2026 - 4:41

score 150 · Answer 13 · 2026-02-23 04:40:32

150

Views

I have written up a proof on why norms on $\mathbb{R}^n$ are equivalent

Published on 23 Feb 2026 - 4:40

score 363 · Answer 14 · 2026-02-23 02:47:18

363

Views

Are metrics uniformly equivalent if and only if they have the same zero-distance sets?

Published on 23 Feb 2026 - 2:47

score 287 · Answer 15 · 2026-02-23 04:39:15

287

Views

Show this induced metric is a metric

Published on 23 Feb 2026 - 4:39

List Question

Show that the metrics $\rho(x,y) = \sqrt{d(x,y)}$ and $\tau(x,y) = \min\{d(x,y),1\}$ are equivalent to $d(x,y)$.

Every two norms on a finite dimensional (real or complex) vector space $V$ are equivalent.

Are d1 and lift metric equivalent distances?

Show that $d$ is equivalent to the usual metric on $\mathbb{N}$ but that $(\mathbb{N},d)$ is not complete.

Prove that two metrics generate the same topology.

Equivalence of Euclidean metric and the metric $d(x,y) = min{1, |x-y|}$ on $\mathbb R^{2}$

Topology induced by equivalent norms

Show that, for every metric $d$, the metrics $d/(1+d)$ and $\min\{1,d\}$ are equivalent

Metric to compare conditional probabilities

Existence of a metric on $\Bbb Q$ which is complete and equivalent to the usual metric.

Determine if two norms are equivalent.

How to prove criterion for topological equivalence of metrics?

I have written up a proof on why norms on $\mathbb{R}^n$ are equivalent

Are metrics uniformly equivalent if and only if they have the same zero-distance sets?

Show this induced metric is a metric

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