Mathematics Stack Exchange
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Questions tagged [equivalent-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, $X$ will denote a non-empty set and $d_{1}$ and $d_{2}$ will denote two metrics on $X$.

(1) The two metrics $d_{1}$ and $d_{2}$ are said to be topologically equivalent if they generate the same topology on $X$.

(2) Two metrics $d_{1}$ and $d_{2}$ are strongly equivalent if and only if there exist positive constants $\alpha $ and $\beta$ such that, for every $x,\ y\in X$, $$ \alpha d_{1}(x,y) \leq d_{2}(x,y) \leq \beta d_{1} (x, y) $$

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Showing equivalent metrics have the same convergent sequences

I am trying to show two equivalent metrics $p$ and $d$ on a set $X$ have the same convergent sequences. $p$ and $d$ are such that $kd(x,y) \leq p(x,y) \leq td(x,y)$ for every $x, y \in X$, $k$ and $t$ are positive constants. Here's what I am doing…
sonicboom
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Equivalent metrics using open balls

Let $d$ and $p$ be two metrics on a set $X$ and let $m$ and $n$ be positive constants such that $md(x,y) \leq p(x,y) \leq nd(x,y)$ for every $x,y \in X$. Show that every open ball for one metric contains an open ball with the same center for the…
sonicboom
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Matrix equivalence independent of dimension

I'm looking for a criterion on symmetric, positive definite matrices $A$ which ensures the constant $M$ in the lower bound of the norm equivalence $M\|A\|_\infty \le \|A\|_2 \le \|A\|_\infty$ does not depend on the dimension of the matrix. EDIT: I…
Plamen
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No equivalent norm induced by inner product

In class I was asked to show that there is no inner product on $\ell^1(\mathbb{N})$ which gives rise to the norm $\|\cdot\|_1$. I was able to do so, using the parallelogram law. Now, I am wondering if it is possible for a norm on…
Todd Burnett
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Equivalent norms without Cauchy-Schwarz inequality

Let $X$ be a finite-dimensional vector space over $\mathbb{F}$. ($\mathbb{R}$ or $\mathbb{C}$) Theorem: All norms on $X$ are equivalent. Proof: $a_k$s and $c_k$s will refer to elements of $\mathbb{F}$. Let $(e_k)_{k=0}^{n-1}$ be a basis of $X$.…
Henricus V.
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About $\rho(x, y)= \begin{cases}\mathrm{d}(x, y) & \mathrm{d}(x, y)<1 \\ 1 & \mathrm{~d}(x, y) \geq 1\end{cases}$ in a metric space

let $(X,d)$ be a metric space now define $\rho(x, y)= \begin{cases}\mathrm{d}(x, y) & \mathrm{d}(x, y)<1 \\ 1 & \mathrm{~d}(x, y) \geq 1\end{cases}$ now which of following options is false ? $(X,d)$ and $(X,\rho)$ have same open sets . $(X,d)$…
amir bahadory
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An equivalent norm in a subspace of $H^2 (\Omega)$

The following questions concerns a problem I am treating in my Masters dissertation. Let $\Omega $ be an open, bounded domain in $\mathbb{R}^3$. Then the norm $$ \Vert u\Vert^2 = \Vert u\Vert_2^2 + \Vert\nabla u\Vert_2^2 + \Vert\Delta u\Vert_2^2…
Danilo Gregorin Afonso
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I have written up a proof on why norms on $\mathbb{R}^n$ are equivalent

Just wanted to share my proof with you smart people to have some feedback and to share each other's ideas. Some disclaimers: this is my first post, English is not my first language, and I know that there are simpler ways to prove this. Here it…
gent96
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When are the sup and euclidean metric interchangeable in analysis on $\mathbb{R}^{n}$

In Analysis on Manifolds, Munkres makes the statement that for most purposes, the sup metric, which is $$max\{|x_{i} - y_{i}| \space i \in \{1 \dots n\}\}$$ and the euclidean metric are "equivalent." In other words, they can be interchanged in…
Muno
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Uniform continuity preserved with equivalent metrics?

I am told that any two metrics that equip a space with the same topology yield the same uniformly continuous functions. Surely this is not true ? The reason I ask is because in one of my exams I'm expected to prove that some function is uniformly…
James Well
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When is a re-scaled metric topologically equivalent to the original metric?

Let $X$ be a metric space with metric $d$, and suppose $d$ is unbounded in the sense that for any $N$ there exist points $x$ and $y$ such that $d(x,y) > N$. To replace $d$ with a bounded metric $d^*$, one might try picking a bounded function $f :…
Rob
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Do two metrics, which define the same convergent sequences also have the same limit?

I am wondering whether the concept of two metrics defining the "same convergent sequences" (for instance, when we are interested in the metrics being topologically equivalent) also includes the respective limits of the sequences. Suppose we have a…
mltm97
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This metrics $d_1(s, t)=\left | {s\over 1 +|s|}- {t\over 1 +|t|}\right| $ and $d_2(s,t)=|s-t|$ ar e equivalents?

In several lists of exercises they affirm that these norms are equivalent: show that this metrics $d_1(s, t)= \left| {s\over 1 +|s|}- {t\over 1 +|t|}\right| $ and $d_2(s,t)=|s-t|$, with $t,s \in \mathbb{R}$, are equivalents. I tried to show that…
leo
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Show that the railway metric is a metric

I'm having trouble proving this. I am able to prove other metrics, I think it is possibly the format of the railway metric that is confusing me... Consider the function $d : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow [0, \infty)$ given…
user712697
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Are metrics uniformly equivalent if and only if they have the same zero-distance sets?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are uniformly continuous. My question is, are two metric…
Keshav Srinivasan
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