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.

learn more… | top users | synonyms (1)

2
votes
0answers
34 views

$A,B$ be countable dense subsets of $\mathbb R$ , let $A,B$ be given usual subspace topologies , then there exists a homeomorphism $f:A \to B$?

Let $A,B$ be countable dense subsets of $\mathbb R$ (with usual euclidean topology ) let $A,B$ be given usual subspace topologies , then is it true that there exists a homeomorphism $f:A \to B$ ?
2
votes
0answers
34 views

Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
2
votes
0answers
39 views

Pseudometric without triangle inequality

I'm working on an optimization problem where I aim to minimize the total euclidean distance of the edges of a graph drawn on a fixed-size grid. For convenience, I actually use the maximum of $0$ and ...
2
votes
0answers
47 views

Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
2
votes
0answers
99 views

The compactness of metric space $\mathcal{G}_n$

Here, metric space $\mathcal{G}_n$ is described below: It is said that it is well-known that $\mathcal{G}_n$ is compact for every $n$. But I can't find a proof, can you give me a proof or an ...
2
votes
0answers
33 views

Function from space of continuous functions to reals is continuous (Proof Verification)

Question: $C$ is the space of continuous functions from $[0,1]$ to $\mathbb{R}$ under the sup metric. Prove the function $$f:C\to\mathbb{R}\quad f\to \int_0^1 f(t)^2 dt$$ is continuous. My answer: ...
2
votes
0answers
43 views

A set is compact iff every collection… Proof check

I asked this question (A set is compact iff all closed collections of subsets with the f.i.p. have nonempty intersection) a few days ago and was lucky enough to get an answer, but I'm afraid that the ...
2
votes
0answers
25 views

$\ell^p$-spaces for $p<1$

It is well known that whenever $p\in (0,1)$, the mapping $$ d_p(x,y):=\|x-y\|_{\ell^p}:=\left(\sum_{n=1}^\infty |x_n-y_n|^p\right)^\frac{1}{p} $$ turns $$\ell^p:=\{(x_n)_{n\in \mathbb N}:\|x\|_{\ell^...
2
votes
0answers
23 views

Limit and Isolation points.

I have attempted a question that says to prove that the set of isolated points of a countable complete metric space X forms a dense subset of X. My Attempt It has been shown previously that the ...
2
votes
0answers
19 views

TVS on the reals which or which not induces convergence in norm

Recently I wondered, if convergence in some given metric $d$ on $\mathbb R^n$ induces convergence in norm. Of course, if $d(x,y) = \|f(x)-f(y)\|$, where $f$ is a bijection on $\mathbb R^n$, then this ...
2
votes
0answers
55 views

How are neighbouring sequences defined? (Metric spaces)

What does it mean for sequences to be neighbours in a metric space? My attempt is: In a metric space $(X,d)$, $(x_n)$ and $(y_n)$ are neighbouring sequences iff $$\forall_{\epsilon>0}\exists_N n\...
2
votes
0answers
22 views

Help understanding a proof that the metric space of bounded functions is complete.

Theorem Let $M$ and $N$ be metric spaces, then the metric space $F_{b}(M,N)$ $= \{f:M \to N : \text{f bounded} \}$ is complete if $N$ is complete Proof Let $\{f_k\}_{k=1}^{\infty}$ be a Cauchy ...
2
votes
0answers
44 views

Proof of the Beltrami theorem

I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem: Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is ...
2
votes
0answers
40 views

Proof of the second property of metrics for metric space $(C^{\infty}[a,b],\rho)$

Let $C^{\infty}[a,b] $ the set of all infinitely differentiable functions on $ [a, b] $, and let for $x,y \in C^{\infty}[a,b]$, $$ \rho(x,y)=\sum_{k=0}^{\infty}\frac{1}{2^k}\cdot \frac{\...
2
votes
0answers
47 views

Let $f$ is a uniformly continuous function on $(0,1).$ Is it possible to approximate $f$ by polynomials.

Let $f$ be a uniformly continuous function on $(0,1)$. Then $f$ can be extended to a continuous function $\widetilde{f}$ on $[0,1]$. By "Weierstrass Approximation Theorem" the extended function $\...
2
votes
0answers
39 views

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 ...
2
votes
0answers
39 views

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 ...
2
votes
0answers
75 views

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 ...
2
votes
0answers
59 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 ...
2
votes
0answers
27 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 $...
2
votes
0answers
16 views

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 ...
2
votes
0answers
14 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 ...
2
votes
0answers
29 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 ...
2
votes
0answers
40 views

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. $f$ ...
2
votes
0answers
42 views

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 \delta_q(u,v)=\left(\int_0^...
2
votes
0answers
58 views

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 ...
2
votes
0answers
57 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 ...
2
votes
0answers
37 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 ...
2
votes
0answers
33 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 ...
2
votes
0answers
33 views

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 ...
2
votes
0answers
97 views

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 ...
2
votes
0answers
40 views

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 $p_{ij}$...
2
votes
0answers
67 views

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 ...
2
votes
0answers
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 : d(x_1,...
2
votes
0answers
101 views

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 ...
2
votes
0answers
86 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 $A \subset X$ (i....
2
votes
0answers
61 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. ...
2
votes
0answers
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 ...
2
votes
0answers
114 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 $z_{n}=\{x_{n},y_{n}\}\...
2
votes
0answers
36 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 $?$
2
votes
0answers
48 views

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 $C[0,b]$. ...
2
votes
0answers
55 views

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}, \...
2
votes
0answers
59 views

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 analysis)...
2
votes
0answers
178 views

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 ...
2
votes
0answers
46 views

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 ...
2
votes
0answers
55 views

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: $$B_{(\mathbb{R}^2,d_{l_2})}((0,0),1) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 <1\}$$ ...
2
votes
0answers
58 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 $$ x=(\mathbf{x},...
2
votes
0answers
63 views

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) \...
2
votes
0answers
32 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 ...
2
votes
0answers
90 views

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 ...