Questions tagged [metric-spaces]
Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space. For questions about Riemannian metrics use the tag (riemannian-geometry) instead.
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Updated Gorelik principle
One version of the Gorelik principle is the following: Let $E,X$ be Banach spaces and suppose $U:E\to X$ is a Lipschitz isomorphism (that is, a Lipschitz bijection whose inverse is also Lipschitz). ...
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Let $X_{n}$ and $Y_{n}$ are two bounded sequence. I need to prove $\sup_{n\in\mathbb N}|X_{n}-Y_{n}|=0$ iff $X_{n}=Y_{n}$ for all $n$
Let $X_{n}$ and $Y_{n}$ are two bounded sequence. I need to prove $\sup_{n\in\mathbb N}|X_{n}-Y_{n}|=0$ iff $X_{n}=Y_{n}$ for all $n$.
I want this proof. Because if you see this in metric space then ...
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Directly proving a metric function is continuous
Let $(X,d)$ be a metric space an $p\in X$ be any point. I want to show that a function $f:X\to\mathbb R$ defined by $f(x)=d(x,p)$ is continuous.
My attempt:
Let $G\subseteq\mathbb R$ be open and ...
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Prove that $d(x_n,x_{n+1})→0$ $\iff$ $(x_n)$ ia a Cauchy sequence.
Let $X\neq\varnothing$ and let $d:X\times X→\mathbb{R}$ be a function such that
$$d(x,y)=0 \iff x=y,$$
$$d(x,y)=d(y,x),$$
$$d(x,z)≤\max\{d(x,y),d(z,y)\}.$$
Let us say that $d$ is a metric. Let $(...
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33
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Surjective continuous Mapping from Compact Metric spaces
let $X$ and $Y$ be compact uncountable metric Spaces. And let $f: X\rightarrow Y$ be a surjective continuous Mapping. Is it True that one can find a closed subset $A$ of $X$, such that $f$ restricted ...
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If $D$ is dense, then for every open $U$ containing $x$, there exists an open $V \in \mathcal{B}$ such that $x \in V \subset U$
I am currently trying to prove something related to dense and openness of sets:
If $D$ is a dense subset of $X$ and for each $n \in \mathbb{N}$, if $\mathcal{B}_n = \left\{B\left(x, \frac{1}{n}\right)...
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33
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Distances in Poincare disk
I think the following result is quite intuitive and ought to be true, but am having difficulty giving a formal proof:
Let $(\mathbb{D}, d_{\mathbb{D}})$ be the unit disk with the Poincare metric, fix $...
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Diameter of small metric balls in a Finsler manifold
Suppose $M$ is a closed manifold with a Finsler metric $F$, and let $d$ be the induced distance on $M$. In general, due to the asymmetry of $F$, the distance $d$ is asymmetric as well.
Consider the ...
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Metrizing pointwise convergence of *sequences* of functionals in a dual space.
Let $X$ be a normed, real vector space of uncountable dimension. Let $X^*$ denote the set of all continuous linear functions from $X$ to $\mathbb{R}$. Does there exist a metric $d : X^* \times X^* \to ...
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What is wrong with my answer: In metric space M, If A is dense in S and B is open in S, prove $ B \subseteq \overline{A \cap B}$
This was a test question. My answer:
B is open in S, so $$B \subseteq \overline{B} \subseteq S \text{(this is what was marked incorrect) }$$
A is dense in S, so $$A \subseteq S \subseteq \overline{A}$$...
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Completeness of Homogenous Triebel and Sobolev spaces
Modern Fourier Analysis by Grafakos states in Proposition 2.3.1:
Let $0 < p,q < \infty$ and $\alpha \in \mathbb{R}$. The homogenous Triebel-Lizorkin space $\dot{F}^{\alpha,q}_p(\mathbb{R}^n) \...
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Metric on the set of non-empty finite subsets (Ex. 2.4 MTH 427/527)
I am taking the course "Introduction to Topology I. General Topology" (MTH 427/527) and stuck on the following exercise (no. 2.4 in the course notes):
Let $S$ be a set and $\mathcal F(S)$ ...
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Do there exist absolutely homogenous metrics that don’t induce a norm?
From the answer to this question, we have that there exist absolutely homogenous, non translation invariant metrics on $\mathbb{R}^2$ that induce a norm. Does there exist a metric space $(\mathbb{R}^n,...
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Are all squared metric distances also divergences?
Let $M$ be a differentiable manifold that is also a metric space $(M,d)$, equipped with the topology induced by the metric distance $d$; further, assume that $d$ is $C^2$ on $M\times M$. Now, let $D:M\...
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continuous function that factors over quotient topology
Given a metric space $(Y,l)$ and a continuous mapping $f:X \mapsto Y$ (w.r.t $(X,\tau_d)$ for some pseudometric $d$) and $R \subset X\times X$ with $a \sim b$ $\Leftrightarrow$ $d(a,b)=0$.
I'm now ...
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Given any compact set in $\mathbb{R}^n$, the closure of the interior of the set is also compact.
In my Advanced Calculus class we had this excercise and I don't know if this proof is correct.
Using the Heine-Borel Theorem, any set A in $\mathbb{R}^n$ is compact iff A it's closed and bounded.
So ...
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Geometric Interpretation of the Gram-Matrix
I wanted to ask what the geometric interpretation of the Gram-Matrix is. In our linear algebra course, we defined it as:
$$G(\vec{v_1}..\vec{v_n}) =
\begin{bmatrix}
\langle\vec{v_1},\vec{v_1}\rangle &...
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Writing the explicit form of antisymmetrising the metric and Ricci tensor
Whilst going through the solutions to a GR worksheet, I struggled to understand a lie in the solutions. The line is:
$g_{\sigma[{\mu}}\nabla_{|\rho|}R_{\nu]}^\rho=\frac{1}{2}(g_{\sigma\mu}\nabla_\rho ...
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Example of a perfect set of irrationals
I'm doing mathematical analysis and I'm stuck with the following question:
Give an example of a nowhere dense closed set of irrational numbers with no isolated points (that is, a perfect set of ...
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If $X$ is a separable and compact metric space, how to prove that unit ball with weak topology is metrizable?
I'm very confused here. This is the problem that I have, but after looking at similar posts on this website I don't think they are the same as what I'm asking (the other posts all refer to the dual ...
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Prove 1/2 is a Lebesgue number of this cover.
Consider the topological space $(X,{\tau_x})$ where X = $(-{\infty}, 2)$ and ${\tau_x}$ is the subspace topology that X inherits from the standard topology of the space of real numbers. Let $(C_n)_{n\...
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Extension of a continuous function $f:X\rightarrow X$ to a function $g: E\rightarrow E$ such that $X$ is embedded in $E$, a Banach space.
We consider a compact metric space $X$ and a continuous function $f:X\rightarrow X$. I know it's possible to embed $X$ into a Banach or separable Hilbert space $E$ using the weak star topology. ...
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Show that the unit sphere in the space of continuous functions $\mathcal{C}_p^0[0,1]$ is not compact
For $p \geq 1$, let $\mathcal{C}_{p}^0 [0,1]$ be the set of continuous functions in $[0,1]$ with the norm:
$\lVert f \rVert_p = \left( \int_0^1 |f(x)|^p dx\right)^{\frac{1}{p}}$
Show that the unit ...
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Equivalence of two metrics on countable product
There is an exercise 9.3#1 from "Topology without tears":
Let $(X_i, d_i), i\in \mathbb{N}$, countable infinity of metric spaces, where every metric is bounded: $\forall X_i\forall a, b \in ...
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Boundary coincidence implies coincidence of centers
In the metric space $(\mathbb{R},|\cdot|)$, if we are given $x,y\in \mathbb{R}$ with the property that for some $r>0$
$$\partial B(x,r) = \partial B(y,r)$$
Then we can conclude that $x=y$.
This is ...
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Comparing two pairs of natural numbers function
Is there a fucntion such that:
$$F: \mathbb{N}^2 \rightarrow \mathbb{N}$$
$$C_1: \mathbb{N}^4 \rightarrow \mathbb{N},\, C_2: \mathbb{N}^4 \rightarrow \mathbb{N}$$
$$
\left.
\begin{matrix}
x_1 - a_x &...
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The Exact Definition of the Skorohod Metric on $\mathcal{D}([0,1],\mathbb{R}^d)$
I understand that the definition of the Skorohod metric for the space, $\mathcal{D}([0,1],\mathbb{R})$, of cadlag functions $[0,1]\to \mathbb{R}$ is given by:
$$
d(x,y):=\sup_{\lambda \in \Lambda}\...
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If every continuous function $f:K \rightarrow \mathbb{R}$ achieves its maximum and minimum on $K$, is $K$ compact? [duplicate]
A well known result from mathematical analysis is that if $K$ is a non empty compact metric space and $f:K \rightarrow \mathbb{R}$ is a continuous function, then $f$ achieves its maximum and minimum ...
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A chain of open covers in a metric space
Let $(X,d)$ be a metric space, and $K\subseteq X$.
Recall: A collection of open sets $\mathcal{G}\subseteq \tau_{X}$ in $X$ is called an open cover for $K$ if the following is true:
$$K\subseteq \...
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Prove that $(x,y)\mapsto\max \{ \vert x \vert, \vert y \vert \}$ is a metric on $\Bbb R^n$ [closed]
I'm having trouble finding proof for triangle inequality on this problem and i would greatly appreciate any kind of help.
proof that function $d(x,y)= \max \{|x|,|y|\}$ to be a metric on $\Bbb R \...
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Is there such a thing as a continuous metric space? [closed]
If one looks at the set of real numbers equipped with absolute value metric, then we notice that any open interval of any size is non empty. I was wondering if continuity of any metric space could be ...
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Exercise 8, Section 3.2 - Metric Spaces: A Companion to Analysis
The question I am trying to solve is the following from Metric Spaces: A Companion to Analysis:
Let $X$ be a complete metric space. Let $A$ be a nonempty set in $X$. Show that $A$ is homeomorphic to ...
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Injective path in path-connected space [duplicate]
In any path-connected metric space, can we connect any two points with an injective path? This seems possible, for instance by "removing" the points where the loop is not injective, but I ...
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For $I:(X,d_1 )\to(X,d_2)$, $I(x)=x$, proving $I$ is continuous iff $\tau_{d_1}=\tau_{d_2}$
Problem.src) Let $d_1$ and $d_2$ two metrics. Let $I : (X, d_1) \to (X, d_2)$ be given by $I(x) = x$. Then is the following true?
$$ \text{$I$ is continuous} \quad\iff\quad \tau_{d_1} = \tau_{d_2} $$
...
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Every probability measure on a metric space is regular
In Billingsley's book Theorem 1 says the following:
Let $S$ be a metric space equipped with the metric $\rho$ and be $\mathcal{S}$ be the corresponding Borel $\sigma$-field. Let $\mu$ be a probability ...
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How can I prove if it's a metric on $\mathbb{R}$? [closed]
I am studying Analysis by my own and I have a problem in the following question.
Suppose $X=\mathbb{R}$ and define the function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0, +\infty)$ as follows:
$$...
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Proof that pointwise continuity implies uniform continuity for functions on compact sets
Theorem: Let $X$ be a compact metric space and $f : X→\mathbb{R}$ be such that $f$ is pointwise continuous for every $x \in X$. Then $f$ is uniformly continuous on $X$.
I was trying to prove this ...
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Homeomorphism between a plane with a taxi-cab metric and railways metric?
Is a plane with a French railways metric homeomorphic to a plane with a taxi-cab metric?
Here are French railways and taxicab definitions
As far as I know, to show homeomorphism, one needs to identify ...
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Series bounded by uniform convergent series is uniform convergent?
I was reading about the Weierstrass $M$-test, and I was wondering about the following:
Suppose that $(f_{n})$ is a sequence of real-valued functions defined on a set $A$, and there exists $(g_{n})_{n ...
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Prove that in a normalized space the function reaches its infimum.
Let X be a normalized space, $x, y \in X$. Function $\varphi:\mathbb{R}→\mathbb{R}$ is given by the formula $\varphi(t)=||x−ty||$. Prove that it reaches its infimum.
$\textbf{My idea:}$ Consider the ...
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Riemannian metric on fixed rank manifold
I know that one can define metrics on the manifold of SPD matrices
$$
\mathcal{S}^n = \{ A \in \mathbb{R}^{n\times n} \ | \ \text{A positive semi-definite} \}
$$
like the Log-Euclidean metric or the ...
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Limiting proportions of a matrix "metric"
This question is based on an earlier posting on the Wikipedia math reference desk, although in that posting the "equals" proportion and the "greater than" proportion were ...
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63
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Nature of metric spaces
Let $X$ be a metric space, and $Y$ be a closed subset of $X$ such that the distance between any two points in $Y$ is at most $1$:
is $Y$ compact ?
is any continuous function $Y\to \mathbb{R}$ bounded?...
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If x1,...,xn are points of a metric s., must there exist a normed s. (E, ||.||) and $y_1,...,y_n\in E$:$d(x_i,x_j )=||y_i-y_j||$ with all i,j=1,...,n? [duplicate]
If x1, x2,...,xn are points of some metric space, does there necessarily exist such a normed space (E, ||.||) and $y_1,y_2,...,y_n\in E$ that $d(x_i,x_j )=||y_i-y_j||$ with all $i,j=1,2,...n$?
I am ...
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If $\lim_{n\to\infty}A_n=A\neq\varnothing$ as set-theoretic limit, is it true that $\lim_{n\to\infty}\mathrm{diam}(A_n)=\mathrm{diam}(A)$?
Let $(X, d)$ be a metric space with topology induced by the metric $d$, and $(A_n)_{n=1}^\infty$ be a sequence of sets in $X$ such that $\lim_{n\to\infty}A_n = A \neq \varnothing$ as a set-theoretic ...
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A Lipschitz map universal with respect to factorization through it of Lipschitz functions.
Let $(X,d)$ be a metric space. Given a metric space $(Y,\rho)$ and a $1$-Lipschitz map $\varphi\colon X\to Y$, we say that a $1$-Lipschitz function $f\colon X\to \mathbb{R}$ factors through $\varphi$ ...
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Exercise 2 Chapter 4.1 - Magnus
I am trying to solve Exercise 2 (Section 4.1.6, Page 129) from Robert Magnus "Metric Spaces: A Companion to Analysis". I have tried to prove item (a), but I am a unsure on how to approach ...
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73
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How to prove that $ \left \{ \prod_{i=1}^{n} U_i: U_i \text{ are open in } X_i \right \} $ is a base for the product topology?
I'm trying to prove that
$$ B = \left\{ \prod_{i=1}^{n} U_i : U_i \text{ are open in } X_i \right\} $$
is a basis of the product topology.
I was trying to use the subbase of the product topology, that ...
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Generalized semi-metrics taking values on a total order - Which spaces admit one?
I have become interested in metrizability theorems and their generalizations to other notions of a metric. Consider an ordered pair $(M,d)$ and a total order $(T,\leq)$ with least element $0$ where $M$...
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Do the set of distances between points need to be bounded for its diameter to be defined?
From Baby Rudin: Let $E$ be a non-empty subset of metric space $X$ and let $S = \{ d(x,y) | x,y \in E \} $
Then the diamater of $E$ is the least upper bound of $S$.
Question: Don't we need that the ...
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