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.

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9 views

Why is $[0, 1/4)$ open in $X = [0, 1]$, and why then is it not open in $\mathbb{R}$?

"Let $X = [0, 1]$ with its usual metric (which it inherits from $\mathbb{R}$). Then the subset $[0, 1/4)$ is an open subset of $X$ (but not of course of $\mathbb{R}$)." Definition: A subset $...
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The lenght of the diagonal of a hypercube of side 1

In $\Bbb {R^n}$ ,how can we find the length of the diagonal of a hypercube of side 1 for the usual known distances : (L2-norm ) or the euclidean distance and (L infinity norm using the max) ?
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Is $f(z)=\frac{|z|^{3/2}e^{i3\arg(z)}-1}{|z|e^{2i\arg(z)}-1}$ bounded for $|z|\in (0,1)$?

Let $f:B(0,1)\setminus\{0\}\to\mathbb{C}$ denote the function defined by $$f(z)=\dfrac{|z|^{3/2}e^{3i\arg(z)}-1}{|z|e^{2i\arg(z)}-1}.$$ Is f bounded above on $B(0,1)\setminus\{0\}$? I believe it ...
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Solving the matrix equation $X-AXB=C$

Let $A\in\mathbb{C}^{p\times p}, B\in\mathbb{C}^{q\times q}, C\in\mathbb{C}^{p\times q}$, if $\|A^n\|\cdot\|B^n\|<1$, then the matrix equation $X-AXB=C$ have a unique solution $X\in\mathbb{C}^{p\...
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Show that the set of differentiable functions is a complete metric space, where $d(f,g)=\max_{x\in[0,1]}\{|f(x)-g(x)|+|f'(x)-g'(x)|\}$

$f$ and $g$ are in $C^1([0,1])$. I can show that the space of functions, along with this metric, is indeed a metric space. But showing that it's complete is proving to be a bit more complicated. Say I'...
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1answer
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Using calculus to show that $f_n(x)=x^n$ is not Cauchy in $C^0[0,1]$

As the title says, I need to prove "using calculus" that the sequence of functions $f_n(x)=x^n$ is not Cauchy in $C^0[0,1]$. The thing that came to my mind is to use the $L_1$ or $L_2$ norm ...
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Proof that $d(x,y) = |\frac{1}{x}-\frac{1}{y}|$ defines a Metric [duplicate]

I am trying to prove that $d(x,y) = |\frac{1}{x}-\frac{1}{y}|$ is a Metric. I already managed to prove: $d(x,y)\geq 0$ $d(x,y)\geq 0$ iff $x=y$ $d(x,y) = d(y,x)$ But I am stuc at the 4th property: $...
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Sequence of open balls

I have this excercise (that I am not sure if it is true): Suppose $K$ is compact and if $\left(B_{j}\right)_{j=1}^{\infty}$ is a sequence of open balls that covers $K$, prove that there is a positive ...
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Prove that $\inf\{t>0: \frac{x}{t} \in C\}$ is a norm [closed]

Consider $\mathbb{R}^n$ with its standard norm $||x||_2= ( \sum |x_i|^2)^{1/2}$. Let $C \subset \mathbb{R}^n$ be a bounded set such that $0 \in int(C)$. Suppose that the following hold: i) $C$ is ...
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Sequence of real numbers that converges in a metric space

Let $(a_n)_{n \geq 0}$ be a sequence of real numbers that converges to $a \in \mathbb{R}$. Let $A = \{ a_n | n \geq 0\}$. Show that a) $\overline{A} = A \cup \{a\}$ b) $A^\circ = \varnothing$ I am ...
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distance between closed and compact set

my professor gave us this excercise: The distance of two subsets $S_{1}$ and $S_{2}$ of a given complete metric space $(X, d)$ : $$ d\left(S_{1}, S_{2}\right)=\inf \left\{d(x, y): x \in S_{1}, y \in ...
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A term for a “general” space in geometry?

Is there a term for what I could call "a geometric space": a pair of a metric and a group of space isomorphisms (preserving distances)?
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Metric Space induced by Pseudometric.

I am getting confused proving this, mainly, because of how the equivalence relation is defined. Moreover, it is defined as $x \sim y$ if $d(x,y) = 0$. My question is, does this mean if $d(x,y) = 0$ ...
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Riemannian Fisher-Rao metric and orthogonal parameter space.

Let $\mathcal{S}$ be a family of probability distributions $\mathcal{P}$ of random variable $\beta$ which is smoothly parametrized by a finite number of real parameters, i.e., $\mathcal{S}=\left\{\...
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There exists points in closure of set whose distance is diameter of set

Assume that diameter of set is defined as $\displaystyle \text{diam}(A) = \sup_{x,y \in A} d(x,y).$ Prove that if $\text{diam}(A) < \infty$ , show that there exists $x',y' \in A^{-}$ s.t. $d(x,y) =...
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1answer
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Coincidence of two topologies

Let $X$ be a metric space. $F$ is a set of mapping from $X$ into itself, Then: If $F$ is equicontinuous on $X$ then on $F$ the topology of pointwise convergence coinsides with the compact-open ...
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1answer
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Topologically equivalent metrics using sequence

Now I am studying about topologically equivalent of two metrics on General Topology. There is an exercise: Given $d$ and $\rho$ are metrics on X which topologically equivalent if only if for an ...
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Pseudo metric space and $T_0$ space

"A pseudo metric space is a metric space iff it's a $T_0$ space" How can I prove this? please help
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If an element is in the closure of $A$, does that mean it is in $A$?

Recently I was reading the proof that the closure of an open ball is contained in the closed ball. It is a simple one-liner at the bottom of this page: http://mathonline.wikidot.com/the-closure-of-an-...
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Is there a name for points of a metric space which are limits of sequences in a subset?

Suppose $X$ is a metric space and $A \subseteq X$. By definition, $x \in X$ is a limit point of $A$ if every punctured ball centered at $x$, $B_r(x) - \{x\}$, contains a point of $A$. This definition ...
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1answer
25 views

Uniform parametrisation of $SO(3)$

Does there exist a map from a hypercube of some dimension to $SO(3)$ such that pushforward measure of standard measure on the hypercube is invariant under action of $SO(3)$ on itself?
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For a given metric space, to show the set $A = \{ p, p_1, p_2, \ldots \}$ is closed if $p_n \rightarrow p$.

The only property of closed sets I am working with here is: A subset $A$ is closed iff the limit of each convergent sequence of points in $A$ is also in $A$. So let $(q_n)_{n\geq 1} = (q_1, q_2, q_3,...
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Possibly differing definitions of local path-connectedness

In C.C.Pugh's book "Real Mathematical Analysis" a metric space $X$ is said to be locally path-connected if for each $p\in X$ and (open) neighborhood $U$ containing $p$ there is a path-...
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37 views

Approximate a function $f$ using continuous function $g$ with respect to $d(f, g) = \int_0^1 |f(x) - g(x)|dx$.

I have the following question: Consider the set of all Riemann integrable functions $[0,1]$ and define the metric $$d(f,g) = \displaystyle\int_{0}^{1}\left|f(x)-g(x)\right|dx\qquad \forall f, g \in ...
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Proving $\sin(x^2)$ Continuous but Not Uniformly Continuous with non-standard Definition

Show that the bounded function $f(x) = \sin(x^2)$ is continuous, but not uniformly continuous, on the interval $[0,\infty)$. The problem should be solved using the following definitions: Definition ...
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15 views

Determinant of the metric tensor

How to calculate or simplify the determinant of the matrix formed by the coefficients of the metric tensor? Since,i am new to this chapter please keep the derivation upto the third dimension.
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Questions about some distance functions

I saw the following written as distance function for random vectors $(\mathbf{X}, \mathbf{Y})$: The Euclidean or $L^2$ distance $\Delta_2 (\mathbf{X}, \mathbf{Y}) = \vert\vert \mathbf{X} - \mathbf{Y} ...
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What can be concluded about $f\ $?

Let $X$ and $Y$ be metric spaces. Assume that $Y$ is a discrete metric space and that $f : X \longrightarrow Y$ is a contraction. What can be concluded about $f\ $? What I observe is that if there ...
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1answer
32 views

Converse of contraction mapping theorem

Let $(X,d)$ be a metric space such that, if $Y \subset X$ is closed, then every contraction mapping on $Y$ has a fixed point. Show that $X$ is complete. This problem appeared in an exam and I ...
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$(1+x)^d \leq 1 + x^d$ for $x \geq 0$ and $d \in (0,1]$

I come across the following inequality: for all $x \geq 0$ and $d \geq 0$, if $d \leq 1$: $$(1+x)^d \leq 1 + x^d,$$ if $d \geq 1$: $$(1+x)^d \geq 1 + x^d.$$ I think they are related to convex ...
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Condition for the norm inequality

Let $\Phi(x)=\int_0^x \phi(y)dy, \, x \in R_+$ is an N-function. Let $u$ is locally inferable on $R_+$. Consider the gauge norm $$ \rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{R_+}\Phi\left(\frac{|f(x)|}...
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Defining a metric on plane based on downward gravity force

What is the metric defined on $\mathbb {R}^2$ plane, where downward gravity force ($\overrightarrow {-y}$ direction) causes the brachistochrone curves become its geodesics?
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1answer
27 views

Inclusion and projection are continuous functions

Let $m \geq n$. Prove that the inclusion $\eta:\mathbb{R}^n \rightarrow \mathbb{R}^m, x \rightarrow (x,0)$ and the projection $\pi:\mathbb{R}^m \rightarrow \mathbb{R}^n, (x,y) \rightarrow x$ are ...
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Is there a notion of measure over arbitrary metric spaces?

The title says it all, but just to be more specific: Given a set $S$ with metric $d$, the measure $m$ would be defined in terms of $d$ and would have properties one would want a measure to have, such ...
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1answer
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$k$-local geodesics are not necessarily quasi-geodesics in flat space.

There is a local to global principle in hyperbolic geometry saying the following: If $X$ is a $\delta$-hyperbolic geodesic space and $k>8\delta,$ then any $k$-local geodesic is a $\lambda, \...
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1answer
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Prove on metrization of uncountable product [duplicate]

I am given the following problem: Show that an uncountable product of unit intervals is not first countable, and thus not metrizable. My answer would be that a), since the elements of the neighborhood ...
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21 views

characterisation of relatively compact sets

Let $X$ be a metric space and $A \subset X$. Why does there holds the following statement: $A$ is relatively compact, i.e. $\overline{A}$ is compact $\Leftrightarrow$ for every sequence $(x_n)_n \...
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(EDITED) Alternative proof of Cauchy Schwarz Inequality implication

The Cauchy Schwarz inequality says that if $x,y\neq0$ there will be equality, that is, $|x||y|=\|x\|\|y\|$ iff for some $c\in \mathbb{R}$, we have $$x=cy$$ (that is $x$ is parallel to $y$) Now, ...
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1answer
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Example of a set that is neither open nor closed in given metric space $M$

I gave this example: $a\in M$, take the closed ball $B[a,r]=\{x\in M | d(a,x)\leq r\}$, take an element $x$ such that $d(a,x)=r$ and do $S=B[a,r]-\{x\}$. It's easy to prove that this set is neither ...
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1answer
45 views

The set of all continuous functions $C(T)$ on a compact pseudo metric space $T$ is a separable.

I'm studying functional-analysis and solving problems in my text book. This is a problem that I cannot solve: $C(T)$ : a set of all continuous functions on $T$ $T$ : a compact pseudometric space ($d(...
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44 views

Given a metric $d$, prove that $\sqrt[n]{d}$ is a metric

I'm having problem with the triangle inequality, since, for $n>2$, it's quite impossible to do the trick of taking $(d(x,y) + d(y,z))^n$.
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Need help with a Real Analysis question [duplicate]

I came across a question from Real Analysis. Here's the question. "Let $C$ be a subset of a compact metric space $(X, d)$. Assume that for every continuous function $h : X\to \mathbb{R}$, the ...
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1answer
32 views

How to use a subset of $\mathbb{N}$ to define an infinite interval in $\mathbb{N}$ (defn. of limits)

Let $(E, d)$ be a metric space and $f:\mathbf{N}\rightarrow E$ a mapping. Consider $\mathbb{N}$ as a subset of the extended real line and note that $\infty$ is a cluster point of $\mathbb{N}$. Suppose ...
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1answer
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A problem on existence of a continuous surjective map from punctured unit disc to closed unit interval

Let D denote the closed unit disc in R^2 . There exists a continuous mapping f : D \ {(0,0)} → {x ∈ |R : |x| ≤ 1} which is onto. I was thinking intuitively that, how we use deformation retraction of ...
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Examples of metric in maplesoft

In the maplesoft website there are examples of metric and killing vectors. What are the name and significance of those geometries ? https://de.maplesoft.com/support/help/Maple/view.aspx?path=...
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A local base for space of probability measures with Prohorov metric

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
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What is the transported metric?

In this previous question: Transported Metric, I wanted to transport the Euclidean metric $ds^2=dx^2+dy^2$ into the first quadrant of the $(u,v)$-plane, such that for all curves $\gamma$ in the $(x,y)$...
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1answer
43 views

Cantor set has dimension zero

Define 'zero-dimensional' as follows: for every pair of closed disjoint sets $A,B$ in a metric space $X$ there exist disjoint open sets $U$ and $V$ such that $A \subseteq U$, $B \subseteq V$ and $X = ...
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1answer
41 views

Open balls with center $\pm\infty$ in the extended real line

Define a function $f$ from $\overline{\mathbb{R}}$ to the interval $[-1,1]$ as follows: For $x\in\mathbb{R}$, let $f(x)=x/(1+|x|)$; on the other hand, write $f(\infty)=1$ and $f(-\infty)=-1$. Then $d(...
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15 views

Problem on fixed point existence

If f is a continuous map from closed unit disc to unit circle, then f must have a fixed point. I was trying that problem, by assuming for shake of contradiction, let f does not have any fixed point , ...

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