Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Uniform implies pointwise convergence

I had a question to show a sequence of functions $(x_n)$ in $C[0,1]$ (equipped with a metric $d$) does not contain a uniformly convergent subsequence. $$ x_n(t) = \ n(1-nt) \ \ \ \ \ \ \forall \ ...
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95 views

is symmetric chi-squared distance “A” metric?

Is symmetric chi squared distance $$\int \frac{(p-q)^2}{pq}\mbox{d}\mu(x)$$ a metric? I am searching web since long time ago but I couldnt find anything. It is positive and is zero whenever $p=q$ ...
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24 views

How does one “separate” the cartesian product properly?

Say, $\delta>0$, $X$ and $Y$ are metric spaces, $(x_0,y_0)\in X \times Y $, and there is some property $P$ such that $$\forall (x,y) \in X \times Y: \ \ \ \ d \Big( (x_0,y_0), (x,y) \Big) < ...
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38 views

The characterization of asymptotic dimension

Let X be a metric space. The following conditions are equivalent (a)asdimX = n (b)n is the smallest integer such that for every R > 0 there exists n + 1 families Ui i=0,1,2,...,n, and S > 0 such ...
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122 views

How can one prove that mahalanobis distance is a metric?

How can one prove that mahalanobis distance is a metric? How can one show that these four properties of a metric are valid for mahalanobis distance? 1) d(x, y) ≥ 0 (non-negativity, or separation ...
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27 views

Proof that compactness can be characterized by closed sets.

If anybody would be willing to check to see if this proof is correct I would really appreciate it. Prove that a metric space $(X,d)$ is compact if and only if for any family $(C_i)_{i \in I}$ of ...
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24 views

space metris disjoint

Let F be a closed subset of a metric space M and p∈M∖F . Show that there are two disjoint open sets G and H in M such that p∈G and H⊆F . I solved well but I think is not the way ... We take a ...
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107 views

Continuity of distance function and its generalization

The starting is an easy undergraduate problem. The distance function $d: X \times X \rightarrow \mathbb{R}$ in a metric space $(X,d)$ is continuous. Please check if my proof is correct. If it is wrong ...
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24 views

Estimating/Reconstructing the distance matrix by given pairwise distance of a subset of points

Given a set of points $X$ which separated into two subsets $X_1$ and $X_2$ i.e. $X_1 \cup X_2 = X$ and $X_1 \cap X_2 = \emptyset$ We have the pairwise distance matrix $M^1$,$M^2$ of set $X_1$, $X_2$ ...
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67 views

lebesgue measure is metric outer measure

this question is driving me crazy.. I need to prove that lebesgue measure is metric outer measure. Unfortunately, I get lost. All what i got is b/c M is LM then M*(A union B) < M*(A)+M*(B) then ...
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184 views

Complete but not totally bounded metric space with a certain property

Give an example of a complete metric space $(X,d)$ and a nested sequence of nonempty closed BALLS $A_n = \bar{B}(x_n,r) = \{y \in X : d(x_n,y) \leq r\}$ such that $\bigcap_n A_n = \emptyset$. So ...
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138 views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
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121 views

Prove equivalent metric spaces

Let $X_1=[1,2]$ and $X_2=[0,1]$. Let $d_1$ denote Euclidean and let $d_2(x,y)=2|x-y|$ in $X_2$. Show that $(X_1,d_1)$ and $(X_2,d_2)$ are equivalent metric spaces. How do I do that?
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47 views

Relationship metric space and $\sigma$-discrete base

Hy, I am newbie here. Can you help me to prove this proposition? If $X$ metric space, then there is a $\sigma$-discrete base $\mathcal{U}$ for the topology of $X$, i.e., ...
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317 views

Showing the Unit Circle is Connected

One way to show that the unit circle is connected is to use the map $f: [0, 2\pi] \to \mathbb{R}^2$ where $f(x) = (\cos x, \sin x)$. Since $f$ is a continuous map and $[0, 2\pi]$ is connected, the ...
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21 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
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65 views

Contraction Mappings

I'm self-learning functional analysis at the moment and although I can understand the underlying theory I have difficulty applying it in the aggregate. Can someone please break this down for me step ...
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194 views

interior, boundary, closure of some sets in plane

$\newcommand{\intr}{\operatorname{int}}$ For the following sets $E ⊆ \mathbb{R}^2 $, I need to find $E'$ , $\overline E$, $\intr(E)$ and $∂E$. (a) $E = \{(x, y) : 1 < x^2 + y ^2 ≤ 4\}$, (b) $E = ...
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51 views

Singular Value Decomposition of Singular Matrices?

$\exists \delta > 0 $ such that whenever $0 < |\varepsilon| < \delta$, $A+\varepsilon I$ is non-singular , for any singular matrix A $\in M_n(\mathbb{C})$ . This is easy to prove. Because ...
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46 views

Computationally efficient means of determining distance in the Skorohod Topology?

I have two functions f and g in a computer. Domain 1...N. I'd like to compute their distance using the Skorohod Topology in an efficient manner. (I first ran across this metric many years ago in ...
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112 views

a problem on metric spaces

I am reading the book by Burago and Ivanov "A course in metric geometry". I tried to do some problems but have some difficulties. For example, page 66 exercise 3.1.26: Let $(X, d)$ be a metric space ...
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22 views

A better way to see this relation concerning Ricci tensor components

If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
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64 views

To show that something is a four-vector

I hope this question is not too inane... it would be really helpful for me to have this cleared up. I want to know what I need to show to demonstrate that something is a four-vector. I have checked ...
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41 views

Hellinger distance between 3-parameter Weibull distributions

I found Wikipedia to have listed Hellinger distance between pairs of 2-parameter Weibull distributions sharing the same shape parameter http://en.wikipedia.org/wiki/Hellinger_distance However, I ...
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103 views

Determining Complete Metric Spaces

I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$ My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
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70 views

Define metric on set and products

Let $X$ be set. My question is: if adding point $\ast$ to $X$ to get set $X \cup \{\ast\}$ then on countable product $\prod_{n \in \mathbb N_+} X \cup \{\ast\}$ I found it possible to define metric. ...
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36 views

Complete metric spaces and specific subsets:

Hello I was given a take home true or false questionnaire and was hoping that you could help me figure out if I am right or not as I was not given the answer sheet. The question is: In a complete ...
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226 views

What is the relation between convex metric spaces and convex sets?

Here's another question that came to mind when I was reading the article on convex metric spaces in Wikipedia: According to the article, "a circle, with the distance between two points measured along ...
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63 views

Fréchet mean for a general shape space

I am posting this question in order to gain a better understand of what the Fréchet mean is for a generalised shape space. So firstly I gather that the Fréchet mean of a probabilty measure $\mu$ on a ...
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97 views

Quotient metric spaces: pseudo metrics versus metrics

I got the following definition from wikipedia: If $M$ is a metric space with metric $d$, and $\sim$ is an equivalence relation on $M$, then we can endow the quotient set $M/{\sim}$ with the ...
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141 views

Showing a differential equation has a unique solution in $C[0, 1]$

Show that $$F(f)(t) = t^2 + \frac{t}{3}f(t) + \frac{1}{5}\int_0^t e^uf(u) du$$ is a contraction on $(C[0, 1), d_u)$. Deduce that the differential equation $$(15 − 5t)\frac{df}{dt} = (5 + 3e^{t})f + ...
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143 views

Should every line be infinite in both two directions?

Yesterday one my classmate asked me for the the definition of line. I have found it in coordinate geometry, in vector space, and in metric space. The definition in metric space is quite general ...
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91 views

Reproducing Kernels are Positive Definite. Does the converse hold true?

Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...
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100 views

what is this called? “difference of the function is less than the function of the difference”

Given: a metric $d$ an aggregate function $f$ some sets (or multisets or random variables) $X$,$Y$ What do we call: $d(f(X),f(Y)) \leq f( [d(X_0,Y_0) \cdots d(X_n,Y_n)] )\ \forall\ ...
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72 views

Reference for the characterization of completeness for metric spaces

I have the following criteria for the completeness of a metric space that I want to use in some research paper. Let $(X,d)$ be a metric space. The following conditions are equivalent: (1) $X$ ...
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114 views

Ruler-and-Compass metric

Let $\nu \left( x \right) $ be the least number of steps that is required to construct a constructible length $x$, using compass and ruler in the well known fashion. Now, define the distance ...
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407 views

Proof of Isometry: Inner Product Preserving Map

For known points $x_i,x_j,\ldots,x_k$, in $\mathbb{R}^n$, consider a mapping $y_i,y_j,\ldots,y_k$ in $\mathbb{R}^n$ produced by minimizing the function $f(y)=\sum_{i,j} \left \langle x_i,x_j \right ...
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85 views

An example of a specific metric space and some functions on it

As part of a research problem my friend asked me to search for a an example of a complete metric space $(X,d)$ and functions $f,g,h$ from $X$ to $X$ such that: (a) Range of $h$ contains the range of ...
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115 views

How to calculate distance of steiner point in Euclidean Steiner Tree?

The euclidean steiner tree for 3 vertices (a,b,c) can be constructed by adding a steiner point (s) connecting 3 edges (as, bs, cs). One way to define the distance of edge (as, bs, cs) is by ...
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Checking for completeness of $\mathbb{R}$ with metric defined by $d_1(x,y) =\mid e^x - e^y \mid$

I have to check for completeness of following metric spaces 1 : $\mathbb{R}$ with metric defined by $d_1(x,y) =\mid e^x - e^y \mid$ for all $x, y \in \mathbb{R}$. 2: $\mathbb{Q}$ with metric ...
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A metric space is path connected and countable then it is complete

I have to show that if a metric space is path connected and countable then it is complete. I'm pretty lost where to start this at all. I have the basic definitions of complete, path-connected, compact ...
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125 views

Show that a subspace $X$ of the Euclidean space $\Bbb{R}^n$ is compact if and only if any sequence of elements of $X$ has a converging subsequence.

Remark: this statement holds in the considerably greater generality of any metric space but the proof of this more general result is quite involved.
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92 views

Distinct metrics on a manifold

I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics ...
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118 views

Showing a set is open with respect to multiple metrics?

I've proven that for $1 \leq p < q \leq \infty$, then $\|x\|_q \leq \|x\|_p \leq n^\frac{1}{p}\|x\|_q$. How can I use this to show that if a set $A\subseteq\mathbb{R}^n$ is open with respect to the ...
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93 views

Finite Levenshtein distance?

Is there a standard term for the relation on sequences where two sequences are related iff they have a finite Levenshtein distance, or for the equivalence classes it induces?
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148 views

l1-metric and cut metric equivalence

I would like to show that the following two statements are equivalent. Let (A, d) be an n-point metric space. And B set of $\binom{n}{2}$ pairs of points of A. $\exists t \geq 1$, integer m, and ...
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211 views

Calculating the Epsilon Neighborhood of line segments in 3d

I am working on a trajectory clustering algorithm (in C++) and one of the steps required in this algorithm is to take a set of 3d line segments (D), and for each line segment (L) in D, to calculate an ...
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10 views

How can the basis elements for the topology induced by the square metric $\rho(x, y)$ be pictured as square regions in the plane?

$\rho(x, y)=\text{max}\{|x_1 - y_1|, |x_2- y_2|\}$. Suppose $x=(0, 0)$ and you were to 'plot' the basis element $\rho(x, y)<1$, how would you do that?
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44 views

On the Gromov-Hausdorff distance

I'm working on my bachelor thesis, and I'm studying principally on two textbooks (Selected Topics on Analysis in Metric Spaces [1] by Luigi Ambrosio and Paolo Tilli and A Course in Metric Geometry [2] ...
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16 views

To show closedness of a subset in a metric spaces

Let $(X, d)$ be a metric space and $p\in X$, $\delta>0$ be fixed. Let $$A=\{q \in X : p \in X, d(p,q)>\delta\}$$ How to show that $A$ is closed? I tried to show that directly by taking $A$'s ...