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.
15,915
questions
1
vote
0
answers
20
views
The existence of $f$ $k$-lipschitz in the subset $Y\subset \mathbb{R}$ implies the existence of a real $k$-lipschitz function $g$ such that $g|_Y=f$.
Here is the problem from a book for metric spaces I'm trying to solve:
Let $f:Y\rightarrow\mathbb{R}$ $k$-lipschitz in the subset $Y\subset \mathbb{R}$. Prove that there is a $k$-lipschitz function $...
- 31
0
votes
0
answers
22
views
when do we say a metric space is quasi-invariant under a function?
A measure of a space that is equivalent to itself under "translations" of this space. More precisely: Let $(X,B)$
be a measurable space (that is, a set $X$
with a distinguished $ σ$
-algebra ...
- 1
-2
votes
0
answers
17
views
Hyperbolic metric space and Cayley graph of a group [closed]
The following definition is given in the book "Group Theory from a Geometrical Viewpoint"
Proposition 2.1. The following are equivalent for a geodesic metric space X.
(1) Triangles are slim....
- 11
7
votes
1
answer
203
views
Problem about fixed points in a complete metric space
Let $(X,d)$ be a non-empty complete metric space and let $ f:X \rightarrow X$ be a function such that for each positive integer $n$ we have
(i) if $ d(x,y)<n+1$ then $d(f(x),f(y))<n$
(ii) if $d(...
0
votes
0
answers
53
views
Prove that the usual metric and other metric induce the same topology
I am working on A course on Borel sets, by S.M. Srivastava. There is this problem I am working on that states the following:
Show that both the metrics $d_1$ and $d_2$ on $\mathbb{R^n}$ defined in 2.1....
- 329
0
votes
0
answers
16
views
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 ...
0
votes
1
answer
55
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?
...
0
votes
0
answers
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: $$$$
Since those ...
- 379
1
vote
2
answers
71
views
Real Analysis Question about Limit points and ε-neighborhoods
The question says "Prove that a point $x$ is a limit point of a set $A$ iff every ε-neighborhood of $x$ intersects $A$ at some point other than $x$."
I am having trouble proving the reverse ...
- 81
0
votes
1
answer
46
views
What, if anything, is this metric on $\mathbb{R}^2$ named? And, what do the open balls in this metric space geometrically look like?
For each $\mathbf{x} := \left( \xi_1, \xi_2 \right) \in \mathbf{R}^2$, let
$$
\lVert \mathbf{x} \rVert := \sqrt{ \xi_1^2 + \xi_2^2 }.
$$
And, for any pair of points $\mathbf{x} := \left( \xi_1, \xi_2 \...
- 26.9k
0
votes
0
answers
56
views
If $\forall n,\sum_ka_{n,k}^2<\infty$ and $\forall k,a_{n,k}\to b_k$, how to show that $\sum_kb_k^2<\infty$? [closed]
Let $\ell^2$ denote the metric space of all the square-summable sequences of real numbers. Let $p_n = \left( a_{n1}, a_{n2}, a_{n3}, \ldots \right)$ for $n = 1, 2, 3, \ldots$ be a sequence of points ...
- 26.9k
0
votes
0
answers
34
views
There is at least one point of every non-empty open subset of the $\ell^2$ space whose first coordinate is nonzero [duplicate]
Here we take
$$
\mathbb{N} := \{ 1, 2, 3, \ldots \}.
$$
Let $\ell^2$ denote the set of all the real (or complex) sequences $\left( \xi_i \right)_{i \in \mathbb{N} }$ such that the series $\sum \left\...
- 26.9k
0
votes
0
answers
69
views
The function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ defined by $d\big((a,b)\big)=\lvert x-y\rvert$ is continuous [duplicate]
Let the function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ be defined by
$$
d\big( (x, y) \big) := \lvert x-y \rvert \qquad \mbox{ for all } (x, y) \in \mathbb{R}^2.
$$
Let $\mathbb{R}$ and $...
- 26.9k
0
votes
0
answers
42
views
The diameter of the union of two sets in a metric space cannot exceed the sum of the diameters of the two sets and the distance between them
Let $A$ and $B$ be any two (nonempty) sets in a metric space $(X, d)$. Then how to show that
$$
d (A \cup B) \leq d(A) + d(B) + d(A, B)? \tag{0}
$$
Here we have the following definitions:
For any (...
- 26.9k
0
votes
0
answers
53
views
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, +\...
- 26.9k
2
votes
2
answers
48
views
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 ...
- 31
0
votes
0
answers
16
views
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$. ...
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 ...
- 358
0
votes
0
answers
42
views
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 ...
3
votes
1
answer
406
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 ...
- 1,131
2
votes
1
answer
75
views
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 ...
0
votes
1
answer
29
views
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\...
- 41
-1
votes
1
answer
18
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\...
- 162
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 $\...
- 59
-1
votes
2
answers
123
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 ...
- 3,075
0
votes
0
answers
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. ...
- 323
0
votes
1
answer
35
views
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 ...
- 477
0
votes
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 ...
- 119
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 ...
- 590
1
vote
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. ...
- 590
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 ...
0
votes
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 ...
0
votes
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 ...
- 119
0
votes
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} \...
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. ...
- 509
0
votes
1
answer
104
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) ...
- 3,044
0
votes
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 ...
- 389
1
vote
0
answers
27
views
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 ...
- 1,020
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 ...
- 6,349
1
vote
1
answer
43
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-...
- 1,020
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 ...
- 2,454
-1
votes
0
answers
24
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 ...
- 579
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 ...
- 199
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 ...
-2
votes
1
answer
101
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^{\...
2
votes
0
answers
33
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 $p=2$ ...
- 477
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 ...
- 199
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_\...
- 1,450
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 ...
- 102
-2
votes
2
answers
21
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 ...