Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from A to B is the same as distance from B to A), positive for two distinct points, and obeying the triangle inequality.
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Finding an example of a bounded sequence in a complete metric space such that the sequence has no partial limit
I'm working through an analysis text and I've come across this exercise:
Give an example of a complete metric space $X$ and a bounded sequence $\left(x_{n}\right)$ in $X$ such that the sequence ...
3
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41 views
Any practical difference between the metrics $d_1=\sup\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ and $d_2=\max\{\left|x_j-y_j\right|:j=1,2,…k\}$?
I've been asked to do a proof showing that $d_1\left({x,y}\right)$ is a metric on $\mathbb{R}^k$, but is there any difference between this and $d_2\left({x,y}\right)$, for which I've already done a ...
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43 views
Metric assuming the value infinity
If we instead define a metric as $d:X\times X \rightarrow [0,\infty]$, do we lose any nice properties of metric spaces?
The reason I ask is that I saw this theorem: Given a finite measure space ...
2
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1answer
30 views
Intuition behind closed subsets of a metric space?
Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space.
Consider a metric space $$(X,d)$$
Then consider a subset of this space$$F$$
What the book ...
4
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1answer
48 views
Continuity in metric space
Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be defined by
$$ F(x,y) = \left( x^3 y,\ \ln(x^2 + y^2 + 1),\ \cos(x - y^2) \right) $$
When trying to show why $F$ is continuous where should I start?
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1answer
85 views
Compact metric space group $Iso(X,d)$ is also compact
Could you tell me how to prove that if metric space $(X,d)$ is compact, then the group $Iso(X,d)$ is also compact?
The group $Iso(X,d)$ is considered with topology determined by a metric $\rho$ on ...
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1answer
50 views
All closed balls are compact each isometry is bijective
Let $(X,d)$ be a metric space in which all closed balls are compact and such that for any two points $x,y \in X$ there exists a function $u \in Iso(X,d)$ such that $u(x)=y$.
Prove that then each ...
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2answers
31 views
Question on Contractions
Let $S = \{x \in \mathbb{R}^n ; \|x\| \le 1 \}$ and $f: S \to S$ be a contraction. Determine one can have $f(S) = S$.
I really need some help with this question. In advance I wanted to give all ...
2
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0answers
30 views
Uniformly continuous function - modulus of continuity
Give an example of a uniformly continuous function $f: (X,d) \rightarrow (Y,\rho)$ for which there doesn't exists a modulus of continuity $\omega: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that:
...
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1answer
46 views
Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.
I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
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1answer
71 views
Square matrix $\|Ax-Ay\|\le \|x-y\|$
Could you give me an example of a square matrix $A\in \mathcal{M}_{2 \times 2}$ or $\mathcal{M}_{3 \times 3}$ for which we have $\|Ax-Ay\|\le \|x-y\|$, $ \ \ x, y \in \{0, e_1, . . . , e_n\}, \ \ e_1, ...
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2answers
63 views
Proving that Euclidean space having the infinity metric is a complete metric space (stuck)
I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space.
I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
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3answers
39 views
Prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ is a metric on $\mathbb{R}^k$
I am trying to prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,...,k\}$ is a metric on $\mathbb{R}^k$. So far I know that $\text{dist}\left({x,y}\right) \leq ...
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49 views
Is any compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?
Every compact metric totally disconnected perfect space is homeomorphic to a Cantor space.
Is every compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?
...
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4answers
31 views
Definition of open set/metric space
On Proof Wiki, the definition of an open set is stated as
Let $(M,d)$ be a metric space and let $U\subset M$, then $U$ is open iff for all $y\in U$, there exists $\epsilon \in \mathbb{R}_{>0}$ ...
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40 views
Example of nested sequence of non-empty closed convex sets in some banach space tha have empty intersection
I know tha if R - banach space, $B_1\supset B_2\supset\dots \supset B_n\supset\dots$ - sequence of nested closed balls in it, then it does have non-empty intersection. But is there an example of ...
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0answers
19 views
How can we define the closedness and boundedness in continuous functions space?
How can we define the closedness and boundedness in continuous functions space?
I'll be very happy if you help me..
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1answer
33 views
How to show C_e is closed and not dense in C.
Let $C_{e}([-1,1],\mathbb{R})$ denote the set of even functions in $C([-1,1],\mathbb{R})$
(a) Show $C_e$ is closed and not dense in $C$.
(b) show the even polynomials are dense in $C_e$, but ...
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2answers
64 views
Show the map $f(x)=\frac12 (x+1/x)$ has an attractive fixed point in $(0,\infty)$
From numerical test, I know $x=1$ is an attractive fixed point of the function
$$
f(x)=\frac12 \left(x+\frac{1}{x}\right),
$$
on $(0,\infty)$.
Is there a way to prove it?
Since
$$
...
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1answer
48 views
Metric and the triangle inequality
Let $A,B$ be finite subsets of the natural numbers. If we let $d(A,B)=\sum_{x\in A\mathbin\Delta B} 2^{-x}$, where $A\mathbin\Delta B=(A\cup B)\setminus (A\cap B)$ is the symmetric difference between ...
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0answers
31 views
A sequence of embedded closed balls that have empty intersection
I'm reading soviet textbook "Elements of theory of functions and functional analysis" by Kolmogorov and Fomin. There is an exercise is in it: show example of complete metric space and a sequence of ...
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1answer
25 views
Hausdorff distance between metric spaces
I started to read a bit Gromov's paper 'Metric Structures for Riemannian and Non-riemannian Spaces'. Tthe Hausdorf distance of two metric spaces $X$ and $Y$ is defined by using isometric embeddings ...
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1answer
45 views
drawing open balls for the radar screen metric
What do the unit balls of this metric look like for radii 1/2, 1 and 3/2 please ?
Radar screen metric is $d(x,y):= \min(1, \|x - y\|_2)$, where the subscript $2$ after the last $\|$ indicates it's ...
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1answer
35 views
For two disjoint compact subsets $A$ and $B$ of a metric space $(X,d)$ show that $d(A,B)>0.$ [duplicate]
I was thinking about the following problem: For two disjoint compact subsets $A$ and $B$ of a metric space $(X,d)$ show that $d(A,B)>0.$ I'm having doubt with my attemp. Please have a look and ...
2
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1answer
48 views
Ergodic theory question about the support of a measure.
I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question:
...
3
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1answer
26 views
How to show that$(X,d)$ is totally bounded
How to show that a metric space $(X,d)$ is totally bounded $\iff$ every infinite subsets of $X$ contains distinct points which distinct points that arbitrarily close to each other.
I don't know how ...
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1answer
63 views
$X \cong Y$, $X$ complete $\implies Y$ complete? [duplicate]
Let $\cong$ denote the homeomorphic notation.
Let $X,Y$ be metric spaces, and let $X \cong Y$. If $X$ is a complete metric space does it imply $Y$ is also complete.
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62 views
Metric on the space of Lipschitz continuous functions
Let $X=C^{[0,1]}([0,1])$, the set of Lipschitz continuous functions with domain $[0,1]$.
a. Prove that
$$\rho(f,g) := \sup|f-g|+\operatorname{Lip}(f-g)$$
is a metric on $X$.
Recall that
...
3
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0answers
67 views
How to show that $\exists~f\in C(X,\mathbb R)$ such that $f$ is unbounded? [duplicate]
Let $X$ be a non-compact metric space. How to show that $\exists~f\in C(X,\mathbb R)$ such that $f$ is unbounded?
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2answers
56 views
Is my proof correct? (minimal distance between compact sets)
I'm working out the following problem form Ahlfors' Complex Analysis text:
"Let $X$ and $Y$ be compact sets in a complete metric space $(S,d)$. Prove that there exist $x \in X,y \in Y$ such that ...
2
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1answer
40 views
(Non) Complete, (Non) separable, (Non) metrizable spaces
After studying several kinds of topological spaces (Like $L_p, C[0,1]$) etc., I thought it would be useful to me (and to others) if I tabulated some of them under 3 categories: Completeness(C), ...
3
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2answers
41 views
Is $D^n$ defined when $n = 0$?
Define $D^n = \{x \in \mathbb{R}^n : |x| \le 1\}$. Is $D^n$ defined when $n = 0$? I would say no, since I don't think $\mathbb{R}^0$ is defined.
3
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2answers
67 views
Intersection of open balls in a metric space
I am wondering about the following question:
Given a (countable) sequence of nested open balls:
$$ B_1 \supseteq B_2 \supseteq \cdots $$
Not necessarily having the same same center. All having ...
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0answers
39 views
Calculate the distance between the points (1, 2, …, n) and (2, 3, … n, 1)
I know that the operation to find the distance between two vectors is:
$$\sqrt{(b_1-a_1)^2+(b_2-a_2)^2+...+(b_n-a_n)^2}$$
So:
The distance between $(7, 5, 3, 1)$ and $(1, 3, 5, 7)$ is:
...
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0answers
15 views
Metric spaces and curvature
Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
3
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1answer
55 views
Do the bounded sequences in any metric space form a complete metric space?
I know that
the set of all bounded sequences over $\mathbb R$ is complete w.r.t. sup norm.
Similarly the set of all bounded sequences over $\mathbb C$ is complete w.r.t. sup norm.
Does this ...
1
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1answer
60 views
Coordinate transform
Can anyone see what transformation $$r\to f(r)$$ transforms $$\exp(2\phi(r))(dr^2+r^2d\theta^2)$$ to
$$df^2+\sinh^2(f)d\theta^2$$?
I there a systematic way to attack such a problem -- rather than just ...
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0answers
19 views
A better way to see this relation concerning Ricci tensor components
If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
3
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2answers
57 views
Metric spaces and distance functions.
I need to provide an example of a space of points X and a distance function d, such that the following properties hold:
X has a countable dense subset
X is uncountably infinite and has only one ...
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2answers
33 views
Which step fails if we would assume $F=(a,b) \subset ℝ$ in the Heine-Borel theorem
I have a question about the theorem: "Every k-cell in $ℝ^k$ is compact". I think it is quite a hard proof, and when I was thinking about it I don't understand the following:
Suppose $F$ is not ...
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0answers
48 views
Showing a Particular Function Between Two Metric Spaces is Continuous
$\fbox{Hypothesis}$
Suppose $(M,d)$ is embedded densely into the two complete metric spaces $(M', d')$ and $(M'', d'')$.
Suppose we define $\rho: M \rightarrow M''$ as the identity mapping.
...
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1answer
52 views
The number of Banach spaces on $\mathbb{R} $
How many possible Banach spaces are there on the entire set $\mathbb{R}$ ?
Thanks
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1answer
37 views
A question on submetrizable space
Let $X$ be a submetrizable space. Then is $X^2$ still submetrizable?
Recall that a space $X$ is called submetrizable if there exists a continuous injection of $X$ into a metrizable space.
Thanks ...
2
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2answers
86 views
$(-\sqrt 2,\sqrt 2)\cap\mathbb Q$ is not compact
How to show that $(-\sqrt 2,\sqrt 2)\cap\mathbb Q$ is not compact in $\mathbb Q$ even though it's closed and bounded in $\mathbb Q$.
Second part is easy. Please help me to show that $(-\sqrt ...
2
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2answers
28 views
A question on Tight probability measures (regular measure)
This is somewhat a basic question, but I'm having difficulty proceeding with a certain part of the proof. I was reading Billingsley "Convergence of Probability Measures", and I encountered the ...
3
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1answer
36 views
Determine if the following is a metric.
I am trying to prove some defined function of two points if it is a metric or not.
From the properties, I am having hard time showing $$\delta (p,q) \le \delta (p,r)+\delta (r,q)$$ for any $p,q,r$ in ...
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0answers
15 views
Parallel transport of a vector along a closed curve in curvilinear coordinates
There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor):
$$
\Delta A_{k} ...
3
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2answers
46 views
Equivalent bounded metric: Why should one prefer $\frac{d}{1+d}$ over $\min\{d,1\}$?
This is my first question and I hope it is not considered too argumentative.
It is often useful to change the metric on a space to an equivalent bounded metric.
Traditionally, people use
$$
...
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1answer
24 views
Relative topology Moore plane
I would like to know how to evaluate the relative topology (If $(X,\tau)$ is a topological space and $A\subseteq X$, then $\tau_A:=\{G\cap A| G\in \tau\}$ is the relative topology) from the following ...
2
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2answers
29 views
Inequivalent metrics can give rise to the same class of Borel sets
I was going through "Convergence of Probability Measures" by Patrick Billingsley. In Section 1: I encountered the following problem:
Show that inequivalent metrics can give rise to the same class of ...
