# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
1answer
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### 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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### 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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### 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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### 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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### 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 ...
1answer
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### $\mathbb N$ 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 ...