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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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Let $f,g\in C[0,1]$ and $U= \{h\in C[0,1]:f(t)<h(t)<g (t),\forall t\in [0, 1]\}$ in $X= (C[0,1], \| .\|_{\infty} ).$ Is $U$ a ball in $X?$

Let $f,g: [0,1]\to\Bbb R$ be continuous and $f(t) < g (t)$ for all $t\in [0,1].$ Consider the set $$U= \{h\in C[0,1] : f (t) < h(t) < g (t) ,\text{ for } t\in [0, 1]\}$$ in the space $X= (C[0,...
Thomas Finley's user avatar
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Can a finite Wasserstein metric on Euclidean support be embedded in a Euclidean space?

Thanks for everyone's help with understanding finite metric embeddings in Euclidean space. I have a follow-up question. Say we have the Wasserstein distance between $n$ distributions in Euclidean ...
user9998990's user avatar
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51 views

Conditions on a finite metric that guarantees embedding in Euclidean space? [duplicate]

If we have $n$ points in some metric space, do there exist coordinates for the $n$ points in an $n-1$ dimensional Euclidean space with exactly the same pairwise distances as in the original space? ...
user9998990's user avatar
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25 views

Proving a the distance between Cauchy sequences converges [duplicate]

Assume we have two Cauchy sequences { $x_n$ } and {$y_n$} in the metric space $(X,d)$. Is it true that the sequence {$a_n$}$=d(x_n,y_n)$ is convergent in $\mathbb{R}$? Here is my try: $$$$ Since those ...
Krum Kutsarov's user avatar
1 vote
2 answers
70 views

Real Analysis Question about Limit points and ε-neighborhoods

The question says "Prove that a point $x$ is a limit point of a set $A$ iff every ε-neighborhood of $x$ intersects $A$ at some point other than $x$." I am having trouble proving the reverse ...
Sachin's user avatar
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What, if anything, is this metric on $\mathbb{R}^2$ named? And, what do the open balls in this metric space geometrically look like?

For each $\mathbf{x} := \left( \xi_1, \xi_2 \right) \in \mathbf{R}^2$, let $$ \lVert \mathbf{x} \rVert := \sqrt{ \xi_1^2 + \xi_2^2 }. $$ And, for any pair of points $\mathbf{x} := \left( \xi_1, \xi_2 \...
Saaqib Mahmood's user avatar
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55 views

If $\forall n,\sum_ka_{n,k}^2<\infty$ and $\forall k,a_{n,k}\to b_k$, how to show that $\sum_kb_k^2<\infty$? [closed]

Let $\ell^2$ denote the metric space of all the square-summable sequences of real numbers. Let $p_n = \left( a_{n1}, a_{n2}, a_{n3}, \ldots \right)$ for $n = 1, 2, 3, \ldots$ be a sequence of points ...
Saaqib Mahmood's user avatar
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There is at least one point of every non-empty open subset of the $\ell^2$ space whose first coordinate is nonzero

Here we take $$ \mathbb{N} := \{ 1, 2, 3, \ldots \}. $$ Let $\ell^2$ denote the set of all the real (or complex) sequences $\left( \xi_i \right)_{i \in \mathbb{N} }$ such that the series $\sum \left\...
Saaqib Mahmood's user avatar
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The function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ defined by $d\big((a,b)\big)=\lvert x-y\rvert$ is continuous [duplicate]

Let the function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ be defined by $$ d\big( (x, y) \big) := \lvert x-y \rvert \qquad \mbox{ for all } (x, y) \in \mathbb{R}^2. $$ Let $\mathbb{R}$ and $...
Saaqib Mahmood's user avatar
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The diameter of the union of two sets in a metric space cannot exceed the sum of the diameters of the two sets and the distance between them

Let $A$ and $B$ be any two (nonempty) sets in a metric space $(X, d)$. Then how to show that $$ d (A \cup B) \leq d(A) + d(B) + d(A, B)? \tag{0} $$ Here we have the following definitions: For any (...
Saaqib Mahmood's user avatar
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Given a metric $d$, is continuity of $f$ absolutely essential for the composite function $f \circ d$ to be a metric (equivalent to $d$)? [closed]

Let $d$ be a metric on a nonempty set $X$, and let $f \colon [0, +\infty) \longrightarrow [0, +\infty)$ be a function such that (i) $f(0) = 0$, (ii) $f(r+s)\leq f(r) + f(s)$ for all $r, s \in [0, +\...
Saaqib Mahmood's user avatar
2 votes
2 answers
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Is there a maximum number of disjoint balls of fixed radius I can fit into a compact metric space?

Let $(X,d)$ be a compact metric space and fix $r>0$. By sequential compactness, one may not find an infinite number of disjoint $r$-balls (sets $B_r(x):=\{y \in X: d(x,y)<r\}$) in $X$ as this ...
Qiyu Xie's user avatar
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Connected Metric Spaces: Strategies

I am not really sure if my ideas in this topic are correct. Can anyone help me? Finding the connected components of a metric space $X$. Suppose there are two connected components $C_1, C_2$ of $X$. ...
Francisco Javier Maciel Hennin's user avatar
1 vote
1 answer
21 views

Convex combination of equidistant curves

Say we have three curves $\gamma, \delta, \varepsilon : \mathbb R \to \mathbb R^n$ such that the distances $\lVert \gamma(t) - \delta(t) \rVert$ and $\lVert \gamma(t) - \varepsilon(t) \rVert$ are ...
markusas's user avatar
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Can any open set in $\mathbb{R}^d$ be countably union of closed sets

I've already know that $\{B(x,r):x\in\mathbb{Q}^d,r\in \mathbb{Q}\} $ is an countable base of $\mathbb{R}^d$. Intuitively, I wonder that can an open set $\Omega\subset \mathbb{R}^d$ be countably union ...
Hải Nguyễn Hoàng's user avatar
3 votes
1 answer
404 views

Is a metric/distance not a measure?

A metric (https://en.wikipedia.org/wiki/Metric_space) or a distance (as in premetric) takes two elements of a set and maps the pair to a real number (or maybe even a complex number). It also might ...
Make42's user avatar
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2 votes
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Finishing the proof of the triangle inequality of Hausdorff metric

currently I am trying to show that a certain type of Hausdorff metric satisfies the following triangle equality and I am stuck. Setup: Take $(X,d)$ as metric space. Denote by $C(X)$ the set of closed ...
a.s. graduate student's user avatar
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How to understand the Sobolev space defined by completion.

In page 16 of this book, the author state: For $1\leq p <\infty$, consider the normed space of all smooth functions $\phi \in \mathbb{R}^n$ such that $$ \|\phi\|_{1,p} = \|\phi\|_p + \|\nabla \phi\...
tianJ's user avatar
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-1 votes
1 answer
17 views

Composition of asymmetric contraction mappings [closed]

Let $(M,d)$ and $(N,q)$ be metric spaces. The operator $T:M\longrightarrow N$ is contractive in the sense that $q(T(m_1),T(m_2)) \leq c d(m_1, m_2)$ for some $c\in [0,1)$. Similarly, the operator $J:N\...
phil's user avatar
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1 vote
1 answer
53 views

Convex cocompact representation of finitely generated groups

Let $\mathbb{H}^n$ be the hyperboloid model for hyperbolic space and $\text{Isom}(\mathbb H^n) = PO(n,1)$. Let $\rho: \Gamma \rightarrow PO(n,1)$ be a representation of finitely generated group $\...
yyffds's user avatar
  • 59
-1 votes
2 answers
122 views

Is my understanding of the definition of a metric space correct?

The definition of metric space that I am using is as follows: Let $X$ be a nonempty set. A function $d:X\times X\to \Bbb R$ is said to be a metric or a distance function on $X$ if $d$ satisfies the ...
Thomas Finley's user avatar
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20 views

Intersection of interiors of sets in a partition of $\mathbb{R}^d$

Let $\mathcal{Q}=\{Q_1,...,Q_n\}$ and $\mathcal{P}=\{P_1,...,P_m\}$ be partitions of $\mathbb{R}^d$ with $n<m$. Assume all $Q_k\in\mathcal{Q}$ and $P_i\in\mathcal{P}$ are close and convex sets. ...
Staltus's user avatar
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1 answer
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Which metrics (on vector spaces) can be induced?

Is there a way to classify which metrics defined on vector spaces can be induced by a norm? ie. there exists norm $n: X\to \mathbb{R}$ on the vector space $X$ such that the metric $d(x,y)=n(x-y)$. I ...
HIH's user avatar
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1 answer
26 views

Axiom of Choice in characterizing openness in subspace

Below is the typical characterization of open sets in a subspace $Y$ of a metric space $X$. $E$ is $Y$-open iff there exists an $X$-open $S$ such that $E = S \cap Y$. The forwards direction usually ...
n1lp0tence's user avatar
1 vote
1 answer
22 views

Does this increasing sequence of subsets of a bounded connected metric space $(X,d)$ terminates at some point at $X$?

This might be a silly question; this is where I'm stuck as to whether a metric-bounded set in a connected metric space is uniformity-bounded in the sense of Bourbaki. Let $(X,d)$ be a bounded ...
Noiril's user avatar
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0 answers
40 views

Is every bounded connected metric space totally bounded?

A metric space $(X,d)$ is said to be bounded if it is equal to a ball $B(x,r)$ of it. It is said to be totally bounded if for all $\epsilon>0$ there is a finite covering of $X$ by $\epsilon$-balls. ...
Noiril's user avatar
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1 vote
1 answer
33 views

Prove equivalent form of Baire's Category Theorem

I'm trying to prove these two statements of Baire's Category Theorem are equivalent: Let X complete metric space. A subset of X is meagre if it can be written as the countable union of nowhere dense ...
Jackson Smith's user avatar
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1 answer
47 views

Are functions from a non metrizable general topological space into $\mathbb{R}$ continous under the ring structure?

That is, would continous functions from a general topological space X be closed under field addition, multiplication and so on? Supposing $X$ is metrizable the proof is pretty doable, and example of ...
Cathartic Encephalopathy's user avatar
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1 answer
36 views

Reconciling metric and topological neighborhoods

Let $X$ be a metric space. Given a point in $x \in X$, an open neighborhood is more appropriately called an $\epsilon$-ball $N_\epsilon = \{p \in X : d(p, x) < \epsilon\}$, while a topological ...
n1lp0tence's user avatar
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1 answer
35 views

If a pseudonorm function $N$ is continuous in a given topology, does the pseudometric topology formed by $d(x,y)=N(x-y)$ coincide with first topology?

Define $p_n\#= p_n p_{n-1} \cdots p_1$ for $n = 0$ to be $1$, then we have a function: $$ N : \Bbb{Z} \to \Bbb{Z}, \\ N(x) = \left |-1/2 + \sum_{d\ \mid\ p_n\#} (-1)^{\omega d} \sum_{r^2 = 1 \mod d} \...
SeekingAMathGeekGirlfriend's user avatar
1 vote
0 answers
33 views

Partition a metric space into parts with small measure and diameter

Consider $\Omega = [0,1]$ with Borel $\sigma$-algebra and Lebesgure measure $\mu$. It has the property that for any $n\geq 1$, we can partition $\Omega$ into $n$ parts with small measure and diameter. ...
efsdfmo12's user avatar
  • 487
0 votes
1 answer
103 views

Do totally open sets exist?

In the third answer to this question, a justification is given for calling closed sets closed, since they are literally closed under the $\mathbb{N}$-ary operation of taking limits of (convergent) ...
Carlyle's user avatar
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3 answers
79 views

Completeness meaning (complete basis vs complete metric space)

Today my professor started talking about the formalism of QM. We talked about the eigenvectors of a Hermitian operator (over Hilbert space) as a "complete set". He also mentioned briefly ...
R24698's user avatar
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0 answers
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Set distances with cube neighbourhoods

This is a follow up and somewhat of a variant of the question I asked a couple of days ago (see Invariance of set distances with $\varepsilon$-neighbourhoods), and after devoting some research and ...
Maths Matador's user avatar
3 votes
1 answer
78 views

Is Hausdorff convergence well behaved with regard to complements of sets?

Let $(X,d)$ be a compact (metric) space and $(A_n)$ a sequence of closed sets in $X$. Let $H$-$\lim_nA_n=A$ (the Hausdorff limit of $(A_n)$). Does $H$-$\lim_n(X\setminus A_n)$ exists? If yes, how is ...
Arian's user avatar
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1 vote
1 answer
42 views

Invariance of set distances with $\varepsilon$-neighbourhoods

I am trying to prove something involving distances between sets which I believe to be true (at least intuitively), but can't seem to get to the end. The situation is as follows. Let $\Omega$ be a non-...
Maths Matador's user avatar
1 vote
1 answer
24 views

Proof of existence of $\epsilon$-nets in infinite metric spaces.

The following is an excerpt from these lecture notes: Given a metric space $(X, d)$, a subset $Y$ of $X$ is said to be an $\epsilon$-net if For $a, b \in Y$, we have $d(a, b) \geq \epsilon$. For all ...
pyridoxal_trigeminus's user avatar
-1 votes
0 answers
23 views

Distances on Cartesian product [duplicate]

I was studying general topology when a question came to my mind. Assume given n metric spaces, call $X$ their Cartesian product and define three real-valued functions from $X\times X$: the first ...
Amanda Wealth's user avatar
2 votes
3 answers
107 views

Proving that the set of limit points of a set is closed directly [duplicate]

I'm working on Baby Rudin chapter 2's exercises and I'm stuck on problem #6, in particular the first part where he asks to prove that the set of limit points E', of a set E, is closed. Here's my ...
user8083's user avatar
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1 vote
1 answer
64 views

Proving that Metric on the set of compact subsets of a metric space inherited from the metric space (Hausdorff metric) satisfies triangle inequality

On a metric space $(M,d)$ and $A \in K(M)$, the set of compact subsets of $M$, we define the function $h: M \times K(M) -> \mathbb{R}_{\geq 0}$, as $h(a,B) = \text{min} \{d(a,b): b \in B\}$, and ...
Cathartic Encephalopathy's user avatar
-2 votes
1 answer
98 views

Convergence of $\sum^{\infty }_{n=0} |a_{n}-b_{n}|$ from convergence of $\sum^{\infty }_{n=0} |a_{n}|$ and $\sum^{\infty }_{n=0} |b_{n}|$

Exercise 12.1.15 of Tao's Analysis II book asks the reader to show that the function $d:X\times X\rightarrow \mathbb{R} \cup \left\{ \infty \right\} $ defined by $d\left( a_{n},b_{n}\right) =\sum^{\...
Diego Martinez's user avatar
2 votes
0 answers
32 views

Linear Analysis – Examples 1-Q5 General $l^p$ spaces vs $l^1, l^\infty$

Let $1<p<\infty$, and let $x$ and $y$ be vectors in $l_p$ with $\left \|x \right \|=\left \|y \right \|=1$ and $\left \|x +y \right \|=2$, how to prove $x=y$? I know how to prove for $p=2$ ...
HIH's user avatar
  • 451
1 vote
3 answers
90 views

Proving that the closure of a set is closed directly

Currently working through Rudin's principle's of mathematical analysis. I am trying to prove directly that the closure of a set is closed but am hitting a wall on one part of the proof. Namely, if we ...
user8083's user avatar
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2 votes
1 answer
55 views

Infinite-dimensional cube $[0,1]^\mathbb N$ compact/complete under certain metrics?

Consider the infinite-dimensional cube $[0,1]^\mathbb N := \{ (x_i)_{i=1}^\infty \ | \ 0 \le x_i \le 1 \ \forall i \}.$ This space can be equipped with certain metrics. Below are two of them. $d_\...
Euclid's user avatar
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1 vote
1 answer
108 views

Determine the convexity of a ball in a metric space.

Let $V$ be the set of all Lebesgue integrable functions. $V$ forms a vector space with respect to general function addition and scalar multiplication. Let $X \subset V$ is the set of positive and ...
Mixi Andrew's user avatar
-2 votes
2 answers
20 views

Proving that boundedness of a metric space defined in terms of radius and diameters are same without usage of triangle inequality

Diameter boundedness:A metric space $M$ is bounded if for all points $p,q \in M$, we have $d(p,q) <= R$ Radius Boundedness: A metric space is bounded if and only if for every point $p \in M$, there ...
Cathartic Encephalopathy's user avatar
1 vote
1 answer
50 views

Two sets having strongly seperated points are strongly seperated themselves

Definition :- Two sets $A$ and $B$ are strongly seperated if there exist open neighbourhoods $U$ and $V$ such that $A \subseteq U , B \subseteq V , U\cap V=\phi$ Question :$A$ and $B$ are two compact ...
user-492177's user avatar
  • 2,589
1 vote
1 answer
47 views

What are the Chebyshev sets for the taxicab metric?

A set of points $S \subseteq \mathbb{R}^n$ is called a Chebyshev set if the metric projection w.r.t $S$ is single-valued. That is, for every point $x\in \mathbb{R}^n$, there is a unique point $y\in S$ ...
Erel Segal-Halevi's user avatar
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0 answers
22 views

Computing Component of Vector Outside Convex Hull of Set of Vectors

I have a set of points $\{x_n\}_{n=1}^N$ where $x_n \in \mathbb{R}^D$. I have another vector $y \in \mathbb{R}^D$. I'd like to efficiently compute the component of $y$ that lives outside the convex ...
Rylan Schaeffer's user avatar
2 votes
0 answers
44 views

Always Closed Metric Space is Complete

I am trying to prove that Given a metric space $Y$, if for every metric space $X \supseteq Y$, $Y$ is $X$-closed, then $Y$ is complete. by contradiction or contraposition, such that I don't use the ...
n1lp0tence's user avatar

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