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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Interchanging limit with infimum/supremum

I'm sure I'm having a notational misunderstanding. Anyway, suppose $(f_n)$ is a sequence of continuous functions from a metric space $X$ to $\mathbb{R}$. So, if $(f_n)$ converges uniformly to a ...
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442 views

quotient metric spaces for dummies

I was hoping that somebody can explain to me the definition of quotient metric spaces I got the following definition from wikipedia: If $M$ is a metric space with metric $d$, and $\sim$ is an ...
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161 views

Existence of minimal subcover for any open cover of a metric space

Suppose (X,d) is a metric space. Does every open cover of X have a minimal subcover with respect to inclusion?
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On $L^p$ and $\ell^p$

If a continuous and infinitely differentiable function $f(x): \mathbb{R}\to\mathbb{C}$ is in $L^p$, is it also true that $f(n),\ n\in \mathbb{Z}$ is in $\ell^p$?
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56 views

Find the most convenient meeting room

In the following question, one has to find the most convenient x-y-z co-ordinates in a building for a group of employees to sit together. I have tried finding individual average values for x, y & ...
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158 views

Help with Metric spaces

$\newcommand{\Int}{\operatorname{Int}}\newcommand{\Bdy}{\operatorname{Bdy}}$ If $A$ and $B$ are sets in a metric space, show that: (note that $\Int$ stands for interior of the set) $\Int (A) \cup ...
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408 views

Algorithms for computing or numerically approximating the Prokhorov metric?

I am interested in the following practical question: Given two measures (say those of two parametric distributions), is there an algorithm for computing the Prokhorov metric between them? The general ...
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185 views

Find Metric in $\mathbb{R}^2$ s.t. it is not Complete

My friend ask me: How to define a metric in $\mathbb{R}^2$ in such an way that $\mathbb{R}^2$ is not complete. I gave him the following metric: Let $B=\{x\in\mathbb{R}^2:\ \|x\|<1\}$. By a ...
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161 views

Why is $L^3$ weaker than $L^2$?

Someone told me today that if I can show $\Vert A_n-B_n\Vert_3\to 0$ as $n\to \infty$, then claiming $A=B$ as $n\to \infty$ (where $A$ and $B$ are the respective limits of $A_n$ and $B_n$) is a weaker ...
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1answer
147 views

Let $L_p$ be the complete, separable space with $p>0$.

Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with ...
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Space with non-convergent Cauchy sequence

Not all sequences that are Cauchy are convergent. Here is what I think the example should be. Somehow the metric space is open but does not contain its limit points. Is this the right direction of ...
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408 views

Triangle inequality for hyperbolic distance

A quick way to define the hyperbolic metric in the Poincare disc is via the cross ratio: Given points a,b in the disc, let p,q be the endpoints of the hyperbolic line (halfcircle/line perpendicular to ...
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2answers
749 views

Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty

The following is problem 3.22 from Rudin's Princples of Mathematical Analysis: Suppose $X$ is a nonempty complete metric space, and $\{G_n\}$ is a sequence of dense open subsets of $X$. Prove ...
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Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
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which of he following is a metric space?

$a)$ $C^1[0,1]$ of continuously differentiable real valued functions on $[0,1]$ with the metric $$d(f,g)=\max_{t\in[0,1]}|f-g|$$ I am sure that it is not complete, but could any one help me to ...
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4answers
698 views

Is $x^n$ Cauchy in $(C[0, 1], ||\cdot||_{\infty})$?

Consider the sequence of functions \begin{equation} f_n(x) = x^n, \quad x \in [0, 1]. \end{equation} Is this sequence Cauchy in $(C[0, 1], ||\cdot||_{\infty})$? The pointwise limit is not ...
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A property of the triangle inequality of Metric Spaces

Question: Assume you have a metric space $(E,d)$ and $A\subset E$ that is nonempty. Define $d_A:x \in E \rightarrow d(x,A)$. Show that $d_A$ is lipschitz and compute its Lipschitz seminorm. My ...
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405 views

Open subsets of a complete metric space.

I've been going over previous exams, and I came across a question that I missed. It is as follows: Let $X$ be a complete metric space. Show that every open subset of $X$ is homeomorphic to a ...
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156 views

A property about open subsets of compact metric spaces.

I've been looking at various problems from past Topology exams, and I came across a problem dealing with compact metric spaces that I have never seen before. The statment to the problem is as follows: ...
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1answer
109 views

Discrete Subgroups of $\mbox{Isom}(X)$ and orbits

Let $X$ be a metric space, and let $G$ be a discrete subgroup of $\mbox{Isom}(X)$ in the compact-open topology. Fix $x \in X$. If $X$ is a proper metric space, it's not hard to show using ...
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Reproducing Kernels are Positive Definite. Does the converse hold true?

Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...
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Proof that $\|(a,b)\| \leq \|(c,d)\|$ if $0 \leq a \leq c$ and $0 \leq b \leq d$ [duplicate]

Possible Duplicate: Is norm non-decreasing in each variable? Let $\| \cdot \|$ be any norm on $\mathbb{R}^{2}$. Let $0 \leq a \leq c$ and $0 \leq b \leq d$. Show that $\|(a,b)\| \leq ...
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A closed noncompact subspace of a metric space

Say that we have a metric space $X$ and that $Y= \{y_1, y_2,\ldots\}$ is a countable collection of points in $X$ such that for any two points in $Y$, we have $d(y_n, y_m) \geq1$, i.e. the distance ...
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216 views

Notation: Representer Theorem for Reproducing kernel hilbert spaces

Am studying the basic concepts of RKHS and the representer theorem: In $f(x_i)=<f,k(x_i,\mathbb{.})>$, what does $ f$ on the r.h.s denote? What is its structure-is it a vector? I was thinking ...
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411 views

Separated sets on a metric space

Let $(X,\rho)$ be a metric space. Two sets $A,B\subseteq X$ are separated if $\overline{A}\cap B=\varnothing$ and $\overline{B}\cap A=\varnothing$. Show that $A$ and $B$ are separated if and only if ...
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291 views

Compact subsets in $l_\infty$ (converse of my last question)

(Converse of my last question) If $A \subseteq \ell_\infty$, and $A=\{l\in \ell_\infty: |l_n| \le b_n \}$, where $b_n$ is a sequence of real, non-negative numbers, then if $\lim (b_n) = 0$ it must ...
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124 views

A metric on $\mathbb{R}^n$ such that $d(\lambda x, \lambda y)=|\lambda| d(x,y)$ which is not induced by a norm

Let $V=\mathbb{R}^n$. Let $d:V \times V\rightarrow \mathbb{R}$ a metric on $\mathbb{R}^n$. Assume that for any $x,y\in V$ and $\lambda \in \mathbb{R}$, we have $d(\lambda x, \lambda y) = ...
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Metrizable compactifications of separable complete metric spaces

I am looking for an example of a separable, completely metrizable space $X$, that has a compactification which is not metrizable. Does such an example exist? And what if $X$ is a separable banach ...
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Prove metric space…

Let $l=\{(a_n)_{n \in ℕ}|a_n \in \Bbb R \text{ and } \sup|a_n|<\infty\}$ If $a=(a_n)$ and $b=(b_n)$ are in $l$, define a metric on $l$ by $$d(a,b)=\sup_{n \in \Bbb N}|a_n-b_n|$$ Prove that $d$ is ...
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How to prove the set $S =\{ (a_1, a_2, a_3)\in \mathbb{R}^3 | \ a_1 + a_3^2\cdot\sin( a_1 +a_2) \geq a_3 \}$ is closed

Let $S = \{ (a_1, a_2, a_3)\in \mathbb{R}^3 | \ a_1 + a_3^2\cdot\sin( a_1 +a_2) \geq a_3 \}$ then, how can I, show that S is closed under Euclidean Metric.
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Compact subsets in $l_\infty$

If $A \subseteq l_\infty$, and $A=\{l\in l_\infty: |l_n| \le b_n \}$, where $b_n$ is a sequence of real, non-negative numbers, then if $A$ is compact subset of $X$ it must mean that $\lim (b_n) = 0$. ...
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1answer
39 views

If $x$ is a limit point of a non empty subset $A$ of a metric space $X$, must it also be a limit point of $Y$ where $A\subset Y\subset X$

If $x$ is a limit point of a non empty subset $A$ of a metric space $X$, must it also be a limit point of $Y$ where $A\subset Y\subset X$? This seems to be trivially true to me. If $x$ is a limit ...
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How to define a “metric” whose range is not the reals?

This may sound a very stupid question. Why do we need to restrict a metric from a general set $X$ to map to the positive real numbers? I try to be clearer. We are given a set $X$ and a totally ...
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1answer
134 views

What is the intersection of this infinite amount of open sets?

Say I have the following $$\bigcap_{n \in \mathbb{N}} \left(-\frac{1}{n}, \frac{1}{n}\right)$$ What is the value of this intersection of an infinite amount of open sets?
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show that $(-1,1)$ and $\mathbb{R}$ are isometric

I want to find a metric $d'$ sucht that $(\mathbb{R}, d')$ and $(X, d)$ with $X = (-1,1)$ and $d(x,y) = |x-y|$ are isometric. I tried the homeomorphisms $$ f:\mathbb{R} \to (-1,1) ~ \textrm{ with } ~ ...
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Metric space, identify E' (real analysis)

The question is: Consider $ \mathbb{R^2} $ with the usual metric and let $E = \{ (t, \sin t) : t > 0 \} $ . Identify $E'$ explicitly. Thank you so much !
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Are these sets in $\mathbb{R}$ open and/or closed?

In $\mathbb{R}$, are these sets open? Are they closed? $A = \{\frac{1}{n} : n \in \mathbb{N}\}$ $B = A \cup \{0\} $ $[0, 1)$ My thoughts: $A$ is not open as if we have an open ball with $r > ...
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1answer
470 views

how to show sets are whether open closed sets

(a) Show that A = {$(x,y) \in \mathbb R^2 : x + y > 5$} is open. (b) Show that B = {$(x,y) \in \mathbb R^2 : x.y \ge 5$} is closed. I couldnt solve in my homework.How should I start this proof? ...
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proving something about the infimum distance between 2 set

Let $S,T$ be 2 set. Prove that there is a $x\in S$ s.t, $d(x,T)=d(S,T)$ if $S$ is a compact set. Here, $d(S,T)$ denoted the $\inf\{d(s,t):s\in S, t\in T\}$.
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Closure of $l_1$ in $l_\infty$

Suppose we have a set $A$ which is the set of all sequences that satisfy $|x_n|\xrightarrow{} 0$. If we consider $l_1$ to be a subset of $l_\infty$. Show that the closure of $l_1$ in $l_\infty$ equals ...
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1answer
248 views

Infimum of Distance in Compact Space

If $X$ is a compact metric space, $A: X\to X$, is it true that if $a = \inf d(x,Ax),\space x \in X$, then there exists $y \in X$ such that $d(y,Ay) = \inf d(x,Ax)$? If so, why?
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Why completion of a metric space $X$ is 'unique' upto isometry?

Let $(X,d)$ be a metric space. Let $({X_1}^*, {d_1}^*)$ and $({X_2}^*, {d_2}^*)$ be completions of $(X,d)$ such that $\phi_1:X\rightarrow {X_1}^*$ and $\phi_2:X\rightarrow {X_2}^*$ are isometries. ...
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98 views

Does a compact subspace have to be closed in an arbitrary metric space?

For Euclidean spaces, we have that a compact subspace has to be closed (and bounded.) But how about an arbitrary metric space? Or how about an arbitrary topology space?
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1answer
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question regarding metric spaces

let X be the surface of the earth for any two points on the earth surface. let d(a,b) be the least time needed to travel from a to b.is this the metric on X? kindly explain each step and logic, ...
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228 views

Two convergent sequences in a metric space.

Question: Let {$x_n$} and {$y_n$} be two convergent sequences in a metric space (E,d). For all $n \in \mathbb{N}$, we defind $z_{2n}=x_n$ and $z_{2n+1}=y_n$. Show that {$z_n$} converges to some $l \in ...
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Prove that the sequence $\{ z_n\}_{n\in \mathbb{N}}$ is cauchy?

I am given the following problem Let (X; d) be a metric space, and let $\{ y_n\}_{n\in \mathbb{N}}$ and $\{ x_n\}_{n\in \mathbb{N}}$ be two sequences in $(X, d)$, both converging towards $a\in ...
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1answer
64 views

Is there an associative (monoid) operation on $\mathbb{R}_{\geq 0}$ which is also a metric?

Is there an associative operation $\star \colon \mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} $ wich also happens to be a metric on $\mathbb{R}_{\geq 0}$? Furthermore is there ...
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4answers
865 views

A problem about topologically equivalent metrics

I tried to solve this problem: Let $(X,d)$ a metric space. Show that $d$ and $\bar{d} =\min({d(x,y),1})$ are topologically equivalent metrics. I proved that $\bar{d}$ is a distance, then I tried to ...
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2answers
625 views

A point in subset $A $of metric space is either limit point or isolated point.

Let $A$ be a subset of a metric space. $A'$ be the set of limit points of $A$ and $A^i$ be the set of isolated points of $A$. Show $A \subset A' \cup A^i$. The picture on my mind is that I draw a ...
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97 views

Every point in a metric space has at least a neighborhood?

I was reading about topology and I came across this statement: Every point $x$ in metric space $(X,d)$ has a neighborhood, which a neighborhood of $x$ (denoted $N(x)$) is defined as there exists ...