Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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Does the metric in the Theory of Relativity actually satisfy the definition of a metric?

Allow me to give a brief introduction to the topic, which has to do with physics; my question will still be a mathematical one. I think my question is aimed at people with a background both in physics ...
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A sequence with no Cauchy subsequence

let $E=\mathcal{C}([0,\pi],\mathbb{R})$ and $$d(f,g)=\sqrt{\int_0^{\pi} (f(x)-g(x))^2 dx}, ~\forall f,g\in E$$ Hello, please How to prove that $f_n(x)=\sin(nx), n\in \mathbb{N}$ has no a Cauchy ...
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Proof of Lemma 19.24 in van der Vaart: step by step proof

In what follows I will sketch the proof of Lemma 19.24 in van der Vaart "Asymptotic Statistics". I'm having troubles in understanding which objects are random and which are not. Thanks in advance for ...
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50 views

Find the interior, closure, and boundary of a given set, in the Euclidean and discrete metrics

I got subset A displayed in image. I'm trying to find Closure, Interior and Boundary in $d^{(2)}$ and $d^{(0)}$ on $\mathbb{R}^2$. Here: $d^{(2)}$ is Euclidean metric, $d^{(0)}$ is discrete ...
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26 views

Completion of subset of $\mathbb{R}^n$

Let $A$ be a metric subspace of $\mathbb{R}^n$ where we have the euclidean metric. How do I prove that $\overline{A}$ with the inclusion map $f:A\hookrightarrow\overline{A}$ is a completion of ...
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Introduction to a textbook on Minkowki spaces

I want to learn more about the metric spaces, specially the "Minkowski spaces" and "Zermelo navigation problem" on Minkowski spaces. I have just studying the book " Riemann-Finsler Geometry" by ...
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12 views

Quantifying the angle metric on the Grassmannian in terms of the norm on the exterior power

Let $V$ be a finite-dimensional Hilbert space and $G_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$. The $k$th exterior power $\bigwedge^k(V)$ can be equipped with a scalar product by ...
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28 views

Proving that union of epsilon nets is an epsilon net

While reading a paper, I came across these definitions and claims: Definition: Given $p \in \mathbb R^d$, and $H$, a set of hyperplanes, let $$\text{Violate}_p(H) = \{h \in H: h \text{ is strictly ...
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Does a mapping from one metric space to another metric space preserve star-likeness of regions?

Let $X$ be a vector space, let $M_1 = \left(X, d_1\right)$ be a metric space and let $M_2 = \left(X, d_2\right)$ be another one. $f : M_1 \to M_2$ is continuous and the origin is a fixed point. $f$ ...
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Distinct topologies on continuous functions $(L_p$ spaces restricted to $C[0,1])$

I am trying to show that the topologies $\tau_p$, $\tau_q$, $\tau_{\infty}$ ($1\leq p<q$) induced by the metrics $$\delta_p(u,v)=\left(\int_0^1 |u-v|^p\right)^{1/p}\quad ...
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Retracts and dense $G_{\delta}$ sets

Suppose $X$ is a complete separable metric space and $Y\subset X$ is a retract of $X$ with retraction $r\colon X\longrightarrow Y$. I am interested in the following question: Given a dense ...
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compact set in a space of functions continuous in $\mathbb{R}$

As it is known the set $\mathcal{B}=\{ f:\mathbb{R}\rightarrow \mathbb{R} : f \mbox{ is continuous}\}$ it is not a metric space with the metric $d(f,g)=sup_{x\in \mathbb{R}}\| f(x)-g(x)\|$ it can ...
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55 views

Limit of Riemannian manifolds is not Riemannian

I want to prove that $D$, standard unit ball in ${\bf R}^2$ with $|\ |$, with a metric $\| \ \|_1$ is a limit of Riemannian manifolds $X_i$. Here problem is to find $X_i$ (If necessary, all metrics ...
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36 views

$W^{1,p}$ is separable for $1\leq p<\infty$

I've been asked to prove that the Sobolev spaces $W^{1,p}(\Omega)$, $\Omega$ open in $\mathbb R^n$, are separable for $1\leq p <\infty$ using the map $$i\colon W^{1,p}(\Omega)\to L^p(\Omega)\times ...
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32 views

Definition of locally connected metric space

I have this definition of locally connected metric space: "A metric space $(X,d)$ is called locally connected if for all $x\in X$ and for all $U\subset X$, $U$ neighbourhood of $x$, exists a connected ...
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Left invariant metrics on a Lie group coming from Lie algebras

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. If we have an inner product on $\mathfrak{g}$ then we can create a left invariant metric $d_G$ on $G$ by translations. On the other hand ...
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Compact convergence of inverse functions

Consider two metric spaces $X$ and $Y$ and a sequence of functions $f_n\colon X\to Y$ together with a function $f\colon X\to Y$. Assume, all $f_n$ and $f$ have inverse functions $g_n$ and $g$, say. It ...
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Find a metric on the simplex so that every transposed positive stochastic matrices becomes a contraction.

A stochastic matrix $P$ is a $n \times n-$matrix with entries $p_{ij} \in [0,1]$ so that $\sum_{k=1}^n p_{ik} = 1$ for every $i \in \{1,...,n\}$. The matrix $P$ is called positive, if no entry ...
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Understanding a theorem of Dynamical systems, mostly definitions

I am reading, the following paper for my research: http://www.sciencedirect.com/science/article/pii/S0022247X00973438 I need to know what are the precise definitions of the following terms, since I ...
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22 views

Proving that a set A in a metric space is bounded iff it belongs to a ball with radius r>0

Let $(X,d)$ be a metric space. Prove that $A\subset X$ is bounded if and only if $A\subset B_r (a)$ for some $a\in X$ and $r>0.$ This is my own proof to the question. Let $B_r(a)=\{x\in X : ...
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Showing the Hausdorff metric inherits completeness

Let $(X,d)$ be a metric space and let $K(X)$ denote the set of all compact subsets of $X$. Then $(K(X), d_H)$ is a metric space, where $d_H$ is the Hausdorff metric. How can I show that if $X$ if ...
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73 views

Image of a precompact under the action of uniformly continuous function is a precompact

Suppose we have two metric spaces $(X, \rho_x)$ and $(Y, \rho_y)$ and a uniformly continuous function $f\colon X \to Y$. The problem is to prove that image $f(A)$ of every precompact $A \subset X$ ...
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59 views

Find a metric on $\mathbb{R}$ with the property that the sequence of natural numbers is Cauchy.

I'm trying to solve the following question: Find a metric $d$ on $\mathbb{R}$ that is equivalent to the usual metric and has the property that the sequence $(n)_{n=1}^{\infty}$ is a Cauchy sequence. ...
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66 views

Problem 13 chapter 4 from baby Rudin

Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function defined on $E$. Prove that $f$ has a continuous extension from $E$ to $X$. Could the range space ...
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42 views

Is there a continuum contained in a compact metric space?

I am reading a paper by M A Armstrong called "the fundamental group of the orbit space of a discontinuous group". In it, he refers to a theorem which says - any light open map between compact metric ...
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101 views

The cartesian product $M\times N$ is complete if, and only if $M$ and $N$ are complete.

The cartesian product $M\times N$ is complete if, and only if $M$ and $N$ are complete. My approach: Let $M$ and $N$, complete metric space, then we take a cauchy sequence ...
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33 views

completion of a complete metric space is itself

Suppose $(X,d)$ is a complete metric space and $(Y,d)$ is its completion . What can be said from this $?$. I think $Y=X$ is the answer . Am I correct $?$
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Proof that the function on $C[0,b]$ is a contraction

Question : Let $a$,$b$ be real numbers with $0\lt b\lt 1$ . Consider the subset $X\subset C[0,b]$ consisting of the functions $f$ s.t $f(0)=a$ .Then $X$ is closed in $C[0,b]$. ...
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Definition of a length metric.

Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \begin{equation} \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, ...
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Proof that the closed interval in $\mathbb{R}$ is connected

Let $C$ be an open and closed subset of $[a,b]$. WLOG, assume $a \in C$. Set $A = \{x \in [a,b]: [a,x] \subseteq C\}$. Since $a \in A$, sup$A$ exists. Let $\epsilon > 0$. Then, (from real ...
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What does it mean when two sets are “adjoined” in a metric space?

I encountered the word "adjoined" in Baby Rudin, Chapter 2 concerning basic topology on Euclidean space. It appeared in the proof to Theorem 2.35 Theorem$\quad$ Closed subsets of compact sets are ...
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Contraction principle for Fredholm integral equation of the first kind

A Fredholm integral equation of the first kind has the following shape: $$ \int a(x,y)f(y)\mathrm dy = b(x)\tag{1} $$ where $f$ is an unknown function. I wonder whether contraction principle can be ...
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Ball with euclidean metric - mistake in book?

In $\mathbb{R}^2$, the ball with euclidean metric $d_{l_2}$ is defined, in terence tao's analysis vol II, as: $$B_{(\mathbb{R}^2,d_{l_2})}((0,0),1) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 <1\}$$ ...
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45 views

Show that $ℓ_2(X)$ is Hilbert space for every set $X$

Show that $ℓ_2(X)$ is Hilbert space for every set $X$ I tryed to find a proof for this problem but i couldn't (searched on internet and mathematical books.Can we find a completed proof for this?
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56 views

Prove or disprove that the Bhattacharyya distance is a true distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ is the space of all symmetric positive-definite $n\times n$ real matrices. Let $x,y\in\mathcal{X}$, where $$ ...
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Hölder's inequality/Cauchy-Schwarz for Bregman Divergence?

Consider the Bregman divergence. $$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle. $$ And its dual norm: $D_{F*}(p, q) $ where $ F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) ...
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26 views

Any compact metric space is Borel equivalent to some subset of $[0, 1]$

In Petersen's Riemannian Geometry book I encountered the following statement : Any compact metric space $X$ is Borel equivalent to some $S \subset [0, 1]$ i.e. there is a bijection $f : X \rightarrow ...
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A subspace of a metric space is normal

Is it true that every subset $Y$ of a metric space $X$ is a normal topological space? I think the answer is yes, because $Y$ is a metric subspace of $X$ equipped with the induced metric by the one of ...
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Can a metric subspace be completely covered by balls after a finite number of steps?

Let $X$ me a metric space with distance $d$ and $A$ be a subspace of $X$. Let $B_\varepsilon(x)$ be the open ball centered in $x$ with radius $\varepsilon$, i.e. $\{y\in X\mid d(x,y) < ...
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Takahashi convex metric spaces

A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies : $d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) ...
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The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...
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Given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected.

Let $X$ be a (metric) space such that given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected. Let us consider a continuous function $f : X \to ...
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Relationsip between two definitions of the christoffel symbol?

When I first started learing about tensor calculus, the professor defined the Christoffel symbol as $$\Gamma ^a _{bc} = Z^a \cdot \partial_b Z_c $$ Where $Z^a$ is a contravariant basis vector and ...
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$A \subset \Bbb R$ such that $A$, $clA$, $int(A)$, $cl(int(A))$, $int(clA)$ are pairwise distinct

Do there exist subsets with internal closures $A$ of $\mathbb R$ such that $A$ , $\bar A$ , $A^\circ$ , $(\bar A)^\circ$ , $\overline{A^\circ}$ are pairwise distinct? I found an example from a book ...
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A metric space of which the geodesic is not a metric

The text book in my course has an exercise about finding a metric space whose (usual) length metric is not a metric. It wants me to find a metric space $(X,d)$ satisfying $d'(x,y)=0 \ \ $for some ...
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A set $U$ is open iff it is union of open balls

Let $(X,d)$ be a metric space. Consider the collection $\mathcal{T} = \{ U \subset X: \forall u \in U, \exists r>0 \; \; , B_r(u) \subset U \}$. We showed that $(X, \mathcal{T} )$ is a topological ...
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118 views

Pointwise convergence imply uniform convergence

I am trying to find a condition under which a sequence of continuous functions on a metric space (or more generally in a topological space) which point wise converge to some function f should imply ...
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118 views

How to show that metrics generate the same topology?

Let $(X, d)$ be a metric space, let $c$ be a positive real number, and define a new metric $d'$ on $X$ by $d'(x,y) = c \cdot d(x,y)$. Prove that $d$ and $d'$ generate the same topology on $X$. ...
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53 views

If $E_i$ is open show $\cap E_i$ is open

Question If $E_i \subseteq \mathbb{R}^p$ is open for all $i=1,2 \dots, n$. Show that $\displaystyle \bigcap_{i=1} ^n E_i$ is open. My attempt: Let $x \in \displaystyle \bigcap_{i =1}^n ...
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36 views

Measuring dispersion

I am trying to define a proper metric for characterizing dispersion of a set of $k \in \mathbb N$ points distributed over different spatial grids. Formally, given different 2-dimensional grids ...