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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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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 ...
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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 ...
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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&...
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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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42 views

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 ...
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26 views

Metric to compare conditional probabilities

I am working on a machine translation task and there are two kind of probabilities: $p_1 = Pr(\epsilon | \phi)$ $p_2 = Pr(\phi | \epsilon)$ Where $\epsilon$ and $\phi$ are phrases in source and ...
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215 views

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

The actual problem looks like- Let, $(X,d)$ be a metric space where $X\ne\emptyset$. Define $d_1$ and $d_2$ on $X\times X$ by $d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}$, $d_2(x,y)=\min\{1, d(x,y)\}\...
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87 views

Topology induced by equivalent norms [duplicate]

Let $X$ be a vector space, equipped with two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ Which are equivalent. What is the easiest way to prove that these two equivalent norms induce same topology?
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Equivalence of Euclidean metric and the metric $d(x,y) = min{1, |x-y|}$ on $\mathbb R^{2}$

I'm trying to prove the following example given in my lecture notes: Example: On $\mathbb R^{2}$ , Euclidean metric and the metric $d(x,y) = \min(1, |x-y|)$ are topologically equivalent. Initial ...
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Prove that two metrics generate the same topology.

Consider the metric $d: X \times X \to \mathbb{R}$, and the metric $\min\{d,1/2\} = d \land 1/2$. Prove that these metrics induce the same topology. Attempt: Let $a \in X, \epsilon > 0$ Then $...
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I don't understand how lp norm of x in R^n is <= (n^1/p) times infinite norm of x !?

I have attached the details in the picture. I don't understand the parts with the red-underlines. Can someone please explain these parts with missing details?
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Show that $d$ is equivalent to the usual metric on $\mathbb{N}$ but that $(\mathbb{N},d)$ is not complete.

Exercise: If we define $$d(m,n) = \left|\dfrac{1}{m} -\dfrac{1}{n}\right|$$ for $m,n\in\mathbb{N}$, show that $d$ is equivalent to the usual metric on $\mathbb{N}$ but that $(\mathbb{N},d)$ is not ...
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Are d1 and lift metric equivalent distances?

I need help proving that $d1\not\equiv d$, where d and d1 are defined as follows: \begin{equation} \label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} d(x,y) = \left\{ \begin{...
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116 views

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

Can someone verify that everything I did in this proof is correct? Every two norms on a finite dimensional (real or complex) vector space $V$ are equivalent. Proof: We assume that $\dim(V)=n$ ...
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Show that the metrics $\rho(x,y) = \sqrt{d(x,y)}$ and $\tau(x,y) = \min\{d(x,y),1\}$ are equivalent to $d(x,y)$.

Exercise: Show that the metrics $\rho(x,y) = \sqrt{d(x,y)}$ and $\tau(x,y) = \min\{d(x,y),1\}$ are equivalent to $d(x,y)$. What I've tried: I know that the metrics $d_0(x,y)$ and $d_1(x,y)$ are ...
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$d_1$ and $d_2$ are equivalent metrics on a set $X$ if and only if there exists a metric space $(Y,\rho)$ and a homeomorphism $h:X\rightarrow Y$

Prove that two metrics $d_1$ and $d_2$ on a set $X$ are equivalent if and only if there exists a metric space $(Y,\rho)$ and a homeomorphism $h:X\rightarrow Y$ from $X$ onto $Y$ such that $d_2(x,y)= \...
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Prove $d$ and $d'=\frac{d}{1+d}$ are equivalent metrics [duplicate]

Suppose $d'(x,y)= \frac{d(x,y)}{1+d(x,y)}$ for $x,y \in X$ and I want to prove $d$ and $d'$ are equivalent metrics on $X$. I would show $$\lim_{n\rightarrow\infty}d(x_n,x)=0 \quad\Longleftrightarrow\...
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Proving the metrics $d_1(x,y) $ and $d_2(x,y) $ are equivalent

Prove that the metrics $$ d_1(x,y) = \sum\limits_{n=1}^\infty \frac{|x_n - y_n|}{ n^{2}(1+|x_n - y_n|)} $$ and $$ d_2(x,y) = \sum\limits_{n=1}^\infty \frac{|x_n - y_n|}{2^{n}(1+|x_n - y_n|)} $$ are ...
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Showing two metrics generate the same topology

I have to show that the metric $e\left(x,y\right)=\frac{d\left(x,y\right)}{1+d\left(x,y\right)}$ and $d\left(x,y\right)$ generate the same topology. I've already shown that $B_{e}\left(x,\frac{\...
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86 views

Metrics on $\mathbb{R}^2$

Definition: We first define three different metrics on $\mathbb R^2$: $(\mathbb{R}^2, d_1)$, $(\mathbb{R}^2, d_2)$ and $(\mathbb{R}^2, d_\infty)$ with $$d_1(x, y):=|x_1 −y_1|+|x_2 −y_2|$$ $$d_2(x, y)...
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108 views

Strong Equivalence of two metrics [duplicate]

Let the two metrics on $X$ be defined as $$d_2(x,y) = \sqrt{(x_1 - y_1)^2 + \dots + (x_n - y_n)^2}$$ $$d_{\infty}(x,y) = \text{max}\left\{|x_1 - y_1|, \dots, |x_n - y_n|\right\}.$$ In the general ...
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660 views

Showing these two metrics are equivalent

Let $(X,D)$ and $(X,d)$ both be metric spaces on the set $X \subset \mathbb{R}^2$ with, $$D(\vec{x},\vec{y})=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$ $$d(\vec{x},\vec{y})=|x_2-x_1| + |y_2-y_1|$$ Show ...
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36 views

Raising metric to a power/equivalent metrics

Suppose we have a locally compact separable bounded metric space $(X,d)$. It is well known that for $\epsilon\in(0,1)$, $d_1(x,y)=d(x,y)^\epsilon$ is a metric. My question is, can we relate $d_1$ and $...
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Prove that $\| \cdot \|_T$ is equivalent to $\| \cdot \|_X.$

Let $(X, \| \cdot \|_X)$ and $(Y, \| \cdot \|_Y)$ be normed vector spaces and assume the map $T \in \mathcal{L}(X,Y)$ is an isomorphism. Define A scalar-valued function $\|\cdot \|_T$ on $X$ by ...
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44 views

Showing that equivalent norms in $\mathbb{R^{n}}$ implies equivalent limits?

In a previous homework, I had to show that $$c_{2}\|x\|_{\alpha}\le\|x\|_{\beta}\le c_{1}\|x\|_{\alpha}$$ where $c_{2}, c_{1}\gt0$ and $\|\cdot\|_{\alpha}$ and $\|\cdot\|_{\beta}$ are arbitrary norms ...
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Understanding the concepts of topologies and equivalent metrics

Could someone please help me in understanding the concepts of topologies and equivalent metrics. If possible, giving some examples of equivalent metrics. For example, I don't know why for the ...
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$||.||_1\leq a||.||_2$. Do the cases $ a∈(0,1) $and $a∈[1,∞)$ matter to be open sets under these norms?

If there are two norms $||.||_1$ and $ ||.||_2 $ on a vector space V such that $||.||_1\leq a||.||_2$ for some positive real number $a$, then can we say that each open set in $(V,||.||_2)$ is also ...
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Defining a metric that quantitatively assesses two sets of data for similarity

I have two sets of data where the x values are identical in both sets. I am trying to determine a quantitative metric to determine how similar the two plots are in terms of shape of the graph, i.e I ...
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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." ...
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Equivalent metrics on $\Bbb N$ [duplicate]

Show that $(\Bbb N,d_1)$ and $(\Bbb N,d_2)$ are equivalent where $d_1$ is the discrete metric and $d_3(m,n)=|\frac{1}{m}-\frac{1}{n}|$. My account In order to show that $(\Bbb N,d_1)$ and $(\Bbb N,...
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48 views

Equivalence of Metrics.

I'm stuck with this problem, I don't even know how to start to work out it: Prove the following metris are equivalent: $$d_1(f,g) = \sup_{N \geq{0} }  \frac{1}{2^N} \frac{\| f-g \|_N}{1 + \| f-g \|...
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179 views

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 ...
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77 views

equivalence of absolute values on number fields

Let $K$ be an algebraic number field such that $[K:\mathbb{Q}]\geq 2$. Let $\sigma:K\rightarrow \mathbb{C}$ be a complex embedding of $K$ ( $\sigma(K)\nsubseteq\mathbb{R}$). Let $\tau:K\rightarrow \...
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197 views

Topology defined with convergence

I can understand why two topologies having the same converging sequences doesn't make them equal. But it must make them similar in some sence to be specified, and if so in what possible ways ? ...
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256 views

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{...
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156 views

Proof of equivalence of $\lambda$ norms in Sobolev space $H_0^1(\Omega)$

Consider the following metrics in $H_0^1(\Omega)$ with $\Omega$ a bounded domain: $$\| u\|_\lambda=\left( \int_\Omega |\nabla u|^2+\lambda\int_\Omega u^2\right)^{\frac{1}{2}}$$ and $$\| u\|_0=\left( \...
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Equivalence of norms on $W^{1, p}(I)$

Let $I=(a, b)$, $a<y<b$ an arbitrary point and $J\subseteq I$ an open interval. Then the norms $\left\Vert \cdot \right\Vert_{\sharp}$, $\left\Vert \cdot \right\Vert_{\flat}$ on the Sobolev ...
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146 views

If $A$ is not totally bounded then $A$ has an infinite subset $B$ homeomorphic to a discrete space

If $A$ is not totally bounded then $A$ has an infinite subset $B$ homeomorphic to a discrete space. My approach since $A$ is not totally bounded we can find $\epsilon>0$ and a sequence $x_n$ ...
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Showing that $\|f\|_1$ is not equivalent to $\|f\|_2$

I have the following norms: $$ \|f\|_1=\int_{t_0}^{t_1}\|f(t)\|_2dt $$ $$ \|f\|_2=\sqrt{\int_{t_0}^{t_1}\|f(t)\|_2^2dt} $$ I need to show their non-equivalence, i.e. that there do not exist numbers $...
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96 views

Equivalence between metrics, show that $d_1\sim d_2\sim d_3$.

Let $d_1,d_2$ and $d_3$ metrics on the metric space $M$. If $d_3\succ d_2\succ d_1$ and $d_1\sim d_3$, then $d_1\sim d_2\sim d_3$. Edit: Where $d_1\succ d_2$, mean that $d_1$ is more fine that $d_2$, ...
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465 views

When are norms not equivalent?

There are a lot of questions here on showing that two norms are not equivalent. I understand that two norms may not be equivaelent from their proofs, however I do not understand why this happened in ...
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725 views

Strongly equivalent metrics are equivalent

I have $d,d'$ metrics in X and that they are strongly equivalent. In my case, this means that: $\exists\alpha,\beta\in\mathbb{R}_{++}$ so that $\alpha d<d'<\beta d$ I want to show that they ...
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85 views

Topologically equivalent metrics, using different definitions.

I´ve been dealing with topologically equivalent metrics for a while, using the usual definition, that $d$ and $d'$ are topologically equivalent iff they have the same open sets. However, there is ...
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64 views

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 ...
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34 views

Is boundedness conserved under equivalent metrics?

Let (X,$\rho$) be a general metric space where $\rho$ is a bounded metric, that is, $\exists M\in\mathbb{R}$ s.t. $\forall x,y\in X$ $\rho(x,y)<M$. Now let $\sigma$ be a metric equivalent to $\rho$....
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260 views

Example of equivalent metrics on the same set such that uniform continuity of some function is not preserved

Give example of a set $X$ and two metrics $d_1,d_2$ on $X$ such that $(X,d_1)$ and $(X,d_2)$ are topologically equivalent but there exist a function $f:X \to X$ which is uniformly $d_1$ continuous but ...
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163 views

$\mathbb N$ when given the metric $d(m,n)=\frac{1}{m}-\frac{1}{n}$

How to show that $\mathbb N$ when given the metric $d(m,n)=\dfrac{1}{m}-\dfrac{1}{n}$ and when given the subspace topology as inherited from $\mathbb R$ are equivalent. When $\mathbb N$ is given the ...