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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.

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Why metric equivalence does not preserve completeness but norm equivalence does? [duplicate]

I was studying Functional analysis when I came across the normed space equivalence i.e. equivalent norms. I already proved that equivalent norms on a vector space $X$ over field $\mathbb{R}$ or $\...
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$(M, g)$ complete Riemannian manifold implies $(M, \tilde{g})$ complete if $\|X\|_{\tilde{g}} \ge \|X\|_g$ for all $X$

The following statement appears as a remark in "An Analytic Criterion for the Completeness of Riemannian Manifolds" by William Gordon (1973): Let $g$ and $\tilde{g}$ be two metric tensors ...
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Equivalence of norms. How do I continue?

I have been working trying to solve a problem. It is about equivalence of norms. I am very close to finishing the problem. I have obtained a lower bound and I still need to obtain the upper bound. I ...
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Comparison between polynomial norms

For $p=\sum_{k=0}^{n}a_k X^k \in \mathbb{C}[X]$, we note $||p||_{\infty}= \max\{|a_k| \: / \: 0 \leq k \leq n \}$ and $||p||_{\mathbb{D}} = \max \{ |p(z)| \: / \: |z| \leq 1 \} = \max \{ |p(z)| \: / \:...
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Prove arctan metric is equivalent to euclidean metric

Given two metrics in $\mathbb{R}$, $d_1(x,y)=|\arctan x-\arctan y|$, and $d_2(x,y)=|x-y|$. I need to prove those two metrics are equivalent. I have the definition that two metrics are said to be ...
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Show that a function is null iff two norms are equivalent

I'm stuck on this question : $Let~E=C(\left [0,1 \right ],\mathbb{R})~and~let~g\in E.~For~all~f\in E~we~set~N(f)=\left\|fg \right\|_{\infty} $ Where $\left\|\phi\right\|_{\infty}=\sup_{t\in \left [ 0,...
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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 ...
Todd Burnett's user avatar
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Necessary and sufficient condition for metric lipschitz equivalence

If $(X,d)$ is a metric space and $f:X \to X$ an injective map, I showed that $(X,d\circ f)$ is also a metric space. And if $f$ is an homeomorphism, then, $d$ and $d\circ f$ are topologically ...
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Equivalence of norms with one boundary condition

There is a paper in which the authors state that, if we consider the Sobolev space $$ \tilde{H}^1 = \{ u \in H^1(0,l)\subseteq\mathbb{R} : u(0) = 0 \} $$ the standard norm $$ \|u\|_{H^1} = \sqrt{\...
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Picard's theorem and Cauchy sequences in different norms

I understand that in order to prove the existence and uniqueness of a solution to a first order non-linear ODE, we first convert it into an integral equation: $y_{n+1} := Ty_{n} = y_{0} + \int_{0}^{t}...
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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 ...
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Non-equivalent norms on $\mathbf{Q}[\sqrt2]$

I found the following post on MO, Can somebody help me filling the details? Just to be sure, here is the restated problem We work on $\mathbf{Q}[\sqrt2]=\left\{x\in\mathbf{R}\mid\: \exists (a,b)\in\...
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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? [duplicate]

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 \...
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Prove that every metric space $(X,d)$ admits a metric $d'$ equivalent to $d$ which makes $X$ bounded.

Exercise Prove that every metric space $(X,d)$ admits a metric $d'$ equivalent to $d$ which makes $X$ bounded. EDIT Definition Let $X$ be a non-empty set. We say the metrics $d_{1}$ and $d_{2}$ ...
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How do prove that $d_{1}$ and $d_{2}$ are equivalent iff the identity map $f:X\to X$ is continuous?

Let $X$ be a non-empty set and $d_{1}$ and $d_{2}$ be metrics on $X$. The following statements are equivalent: (a) $d_{1}$ and $d_{2}$ are equivalent. (b) $f:X\to X$ given by $f(x) = x$ is continuous ...
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Prove $\frac{1}{\sqrt{n}}\|A\|_{\infty} \leq\|A\|_{2} \leq \sqrt{m}\|A\|_{\infty} $

I was trying to prove the following inequality, where $A \in \mathbb{C}^{m\times n}$. $$\frac{1}{\sqrt{n}}\|A\|_{\infty} \leq\|A\|_{2} \leq \sqrt{m}\|A\|_{\infty} $$ But I was really having ...
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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 ?...
amir bahadory's user avatar
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Euclidean metric vs Taxicab Metric in R2

I am wondering when can the Taxicab metric be less than the Euclidean metric. I am aware that usually the Taxicab metric > Euclidean metric and it would be the same if the 2 set of points are in ...
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Strongly equivalence and isometric spaces

I have the following definitions Definition; Two metric spaces $(X,d_1)$ and $(X,d_2)$ are said to be isometric if there is bijective isometry from one space to other. Definition:Two metric spaces $(X,...
Ibrahim Islam's user avatar
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a question about a non-atomic probability space

I am reading "Mathematical Foundations of the Calculus of Probability" by J. Neveu (the English translation). There is the following exercise with hints which I don't quite understand: Say ...
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Property of equivalent measures

Say I have a probability space ($\Omega, \mathcal{F}$) with two equivalent measures $\mathbb{P}$ and $\mathbb{Q}$. Consider two sets $A, B \in \Omega$. I know that $\mathbb{P}\left[A\right] = \mathbb{...
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Spherical metric restricted to the complex plane is equivalent to the usual Euclidean metric.

Let $a \in \mathbb C$ and $r \gt 0$ be given; then there exists $\rho \gt 0$ such that $B_{\infty} (a,\rho) \subseteq B(a,r),$ where $$B_{\infty} (a,\rho) = \{z \in \mathbb C\ |\ d_{\infty} (z,a) \lt \...
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Existence of equivalent norm

Let $L$ be a normed space with norm $\|\cdot\|_1: L\to \mathbf{R}$. Let $T$ be a linear invertible operator on $L$ such that $\|T^n x\|_1 < c \|x\|_1$ for all $x\in L$ and $n\in \mathbf{N}$. Show ...
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Topologically equivalent and lipschitz equivalent. [duplicate]

I am looking for an example of a metric that is topologically equivalent but not lipschitz equivalent. Can you help me? I would be very happy if you can give such an example. I thought a lot but could ...
Dilara Yavuz's user avatar
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Equivalent metrics in $\ell ^\infty$ space

$$\ell^{\infty}=\{z=(z_n):{\sup}_{n}|z_n|<\infty\} \ and\ x,y\in \ell^{\infty}$$ $$d_{\infty}(x,y)=sup_{n}|x_n-y_n| $$ and $$d(x,y)=\sum_{n=1}^{\infty}{\frac{1}{2^n}.{\frac{|x_n-y_n|}{1+|x_n-y_n|}}}...
Anıl Geyik's user avatar
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1 answer
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Does norm equivalence preserve the reflexivity of Banach Spaces?

I am working on a problem right where I have a sequence $(x_n)$ which converges weakly to some $x$ in a Hilbert space $(H, \left<\cdot, \cdot\right>)$. Denote by $||\cdot||$ the norm induced by ...
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Equivalent norms on $\Bbb{R}^n$

I just want my proof checked that all norms on $\mathbb{R}$ are equivalent to the Euclidean norm. It's as follows: Let $||\cdot || $be a norm on $\mathbb{R}^n .\ $Define $f:S^{n-1}\rightarrow \mathbb{...
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Topologically equivalent metrics using sequence

Now I am studying about topologically equivalent of two metrics on General Topology. There is an exercise: Given $d$ and $\rho$ are metrics on X which topologically equivalent if only if for an ...
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Weak convergence in two $L^2(I, \mathbb{R}^n)$ spaces with (bi-Lipschitz) equivalent induced norms

Let $\|\cdot\|_{\mathbb{R}^n}$ denote the Euclidean norm on $\mathbb{R}^n$ and let $g: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be an inner product on $\mathbb{R}^n$ such that: $$\alpha \|x\|_{...
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Are these two metrics equivalent?

Let $d_1$ and $d_2$ be metrics on the space $X$. Assume that for any sequence $\{x_n\}_{n=1}^\infty \subset X$ and point $x_0 \in X$ we have that $$ \lim_{n \to \infty}d_1(x_n,x_0)=0 \iff \lim_{n \to \...
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Relation between spectral radius when the norms are equivalent

If two operator norms are equivalent on B(X), set of all bounded operators on a Banach space X, whether the corresponding spectral radii are the same? If so, please provide proof or any hint. If not, ...
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How do we prove the metrics $d_{j}:\textbf{R}^{n}\times\textbf{R}^{n}\to\textbf{R}_{\geq0}$ are equivalent

Given two points $x,y\in\textbf{R}^{n}$, prove the following inequalities \begin{align*} d_{2}(x,y) \leq d_{1}(x,y) \leq \sqrt{n}d_{2}(x,y) \end{align*} MY ATTEMPT According to its corresponding ...
user0102's user avatar
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If $(X,\|\cdot\|_1)$ is strictly convex, then $(X,\|\cdot\|_2)$ is NOT ALWAYS strictly convex, where $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent.

A normed vector space $(X,\|\cdot\|)$ is strictly convex if and only if $x \neq 0$ and $y \neq 0$ and $\|x+y\|=\|x\|+\|y\|$ together imply that $x=cy$ for some constant $c>0$. We have equivalent ...
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Show that if the norms $∥⋅∥_1$ and $∥⋅∥_2$ are equivalent, then if space $(X,∥⋅∥_1)$ is complete, then the space $(X,∥⋅∥_2)$ is complete.

$X$ is complete if Cauchy sequence converges to its limit in $X$ $$\forall \epsilon > 0,\ \exists N:\ \forall n > N:\ \lVert x_n - x \rVert_1 < \epsilon$$ So let $(X, \lVert \cdot \rVert_1)$ ...
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homeomorphism between an open sphere and $\mathbb {R}^{n}$

Thinking about the fact that $f:\left(\frac{-\pi}{2},\frac{\pi}{2}\right) \rightarrow \mathbb{R}$ where $f(x)=\tan(x)$ for all x in $\operatorname{dom}(f)$ is a homeomorphism between $\left(\frac{-\pi}...
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Prove $d_1(x,y)=|x-y|$ and $d_2(x,y)=|\frac{1}{x}-\frac{1}{y}|$ are equivalent metrics

I am trying to prove whether $d_1(x,y)=|x-y|$ and $d_2(x,y)=|\frac{1}{x}-\frac{1}{y}|$ are equivalent metrics or not. Both metrics are defined in the same space $E=\mathbb{R}^+$. We can check they ...
Gibbs's user avatar
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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 + ...
Danilo Gregorin Afonso's user avatar
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Show that the railway metric is a metric [duplicate]

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 \...
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Find $r_2$ such that $U_1(x, r_2) \subset U_2(x, \varepsilon)$ to show two metrics are equivalent.

Consider $X := (0, 1]$ and the metrics $d_1(x, y) := \left| \frac{1}{x} - \frac{1}{y} \right|$ and $d_2(x, y) = |x - y|$ and show they are topologically equivalent, i.e. for all $x \in X$ $$ \forall ...
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Characterisation of equivalent metrics

Definition (equivalent metrics) Two metrics on a set $X$ are called equivalent if for all $x \in X$ $$ \forall \varepsilon > 0 \ \exists r > 0: U_1(x,r) \subset U_2(x,\varepsilon) \quad\text{...
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Proving that $\rho(x,y) = \frac{d(x,y)}{1+d(x,y)}$ is a metric and that $\rho(x,y)$ and $d(x,y)$ are equivalent metrics [duplicate]

As you can see the proof is divided into 2, the first part consists on proving that $\rho(x,y)$ is a metric My attempt i) $\rho(x,y) \geq 0$, which is clear since $d(x,y)$ is a metric, and it is $0$ ...
Fer Stein's user avatar
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Possible metrics on the space $C[0,1]$?

I had a question that what are all the possible metrics on the space $C[0,1]$ of all continuous functions on $[0,1]$. This question arose when I was trying to prove that $$\mathscr{F} := \left\{F(x) =...
Rick's user avatar
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Two compatible topologically equivalent norms on a module over a non-archimedean field but not metrically equivalent?

The direct question is: Let $F$ be a field, with the trivial norm, that is, $|x|=\begin{cases}1&x\ne0\\0&x=0\end{cases}$. Let $M$ be a free module over $F$ with basis $\left\{x_i\mid i=1,2,...
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Trivial norm is equivalent only to itself

Trivial norm is equivalent only to itself (on a field) The norm here fulfills $3$ requirements $(1) |x|\ge0$ and $ |x|=0\iff x=0$ $(2)|x\cdot y|=|x|\cdot|y|$ $(3)|x+y|\le|x|+|y|$ $2$ norms are ...
Jno's user avatar
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3 answers
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Show this induced metric is a metric [duplicate]

Let $d$ be a metric on $X$ and $(X,d)$ a non-compact metric space i.e. $X = \mathbb{R}$ or $\mathbb{Q}$ For $x,y \in X$ define $$ \tilde{d}(x,y) := \begin{cases} d(x,y), & \text{if }d(x,y) <...
ViktorStein's user avatar
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3 votes
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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 ...
Keshav Srinivasan's user avatar
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1 answer
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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 ...
gent96's user avatar
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How to prove criterion for topological equivalence of metrics?

I have problems proving the following statement. Prove that two metrics$\quad d_1,d_2 $ are topologically equivalent if and only if $$\forall x\in E\quad \forall \epsilon>0 \quad\exists \delta&...
Emerald's user avatar
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2 votes
2 answers
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Determine if two norms are equivalent.

In the space of functions $\mathcal{C}^1([0, 1])$ we define two norms: $$N_1 = \bigg( \int_0^1 |f(x)|^2dx \bigg)^{1/2} + \bigg( \int_0^1 |f'(x)|^2dx \bigg)^{1/2} \\ N_2 = |f(0)| + \bigg( \int_0^1 |f'(...
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Existence of a metric on $\Bbb Q$ which is complete and equivalent to the usual metric.

Does there exist a metric $d$ on $\Bbb Q$ which is equivalent to the usual metric on $\Bbb Q$ such that $(\Bbb Q,d)$ is complete? I have a confusion regarding that. Because I know that equivalence of ...
little o's user avatar
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