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.

learn more… | top users | synonyms (1)

1
vote
1answer
35 views

Metrizable compact spaces and Hausdorff spaces with a countable network

I have two questions related to metrizable spaces and countable network ; Can we find a continuous mapping from a separable metric space onto a non metrizable compact Hausdorff space. If a Hausdorff ...
2
votes
1answer
71 views

Reiterate Volterra integral operator is a contraction

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 472 here) the statement that Volterra operator $A:L_2[a,b]\to L_2[a,b]$ defined ...
-1
votes
2answers
40 views

Is the cardinality of the set of all isolated points in a second countable metric space at-most $\aleph_0$?

Is the cardinality of the set of all isolated points in a second countable metric space at-most $\aleph_0$ ?
1
vote
1answer
61 views

Prove that the product space is a metric space.

I have the following problem: Let $(S,d)$ and $(T,e)$ be two metric spaces. Their product space has underlying set $$S\times T=\{(s,t)|s\in S,t\in T\}$$ and metric ...
0
votes
1answer
23 views

Scalar Products equation proof

$\langle \langle x + y, z \rangle \rangle = \langle \langle x, z \rangle \rangle + \langle \langle y, z \rangle \rangle$ It is clear when there are only $\langle \dot \ , \dot \ \rangle$ but what ...
1
vote
1answer
47 views

Proving a subspace of $l^1_\infty$ is compact

Any help on this would be appreciated. I'm trying to prove that the subspace $(E,\rho)$ is compact. $$E = \{\{x_n\}_n \in X: |x_n|\leq1/(3^n)\text{ for every }n\}$$ $$X=\{\{x_n\}_n \in X: \sum ...
0
votes
0answers
60 views

Is metric always quadratic?

The derivation of metric tensor relies on infinitesimal displacements, which relies on Pythegoras theorem (a specific case of cosine law) to deduce that infinitesimal distance are related to the ...
1
vote
0answers
25 views

$d,e$ be metrices on $X$ , under what condition(s) the function $g:X \times X \to \mathbb R $ , $g(x,y):=\min \{d(x,y),e(x,y)\}$ , is a metric ?

Let $d,e$ be metrices on a set $X$ , then under what condition(s) the function $g:X \times X \to \mathbb R $ defined by $g(x,y):=\min \{d(x,y),e(x,y)\}$ , is a metric ?
2
votes
1answer
40 views

Under what condition(s) is the set of all isolated point in a 2nd countable metric space is empty? [closed]

In what condition the set of all isolated point in a 2nd countable metric space is empty? $NOTE :-$ Sir Brian Scott's answer is ok , but I would like an answer of this edited form , Thanks in ...
0
votes
1answer
25 views

Show Open/Closed for a Set and two continuous functions

Let f, g : X → R be two continuous functions defined on a metric space X. (i) Show that the set U = {x ∈ X : f(x) > g(x)} is open in X. (ii) Show that the set F = {x ∈ X : f(x) ≥ g(x)} is closed in X. ...
3
votes
3answers
47 views

Why does the support of measure on $\mathbb{R}^n$ exist?

DEFINITION : The support of a measure on $\mathbb{R}^n$, written spt $\mu$, is the smallest closed set such that $\mu(\mathbb{R}^n \setminus X)=0$. Why does this set exist?
0
votes
1answer
21 views

How to get a valid distance metric?

I have got a problem to devise a distance metric to get the similarity measurement of vectors. Someone suggested me to use dot product, which seems to me the same as the Cosine similarity metric; ...
1
vote
0answers
11 views

Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
1
vote
2answers
64 views

Show that $\lim_{p→∞} ||x||_p = ||x||_∞$

For any $x ∈ \mathbb{R}^n$ and $p ≥ 1$, define $$||x||_p = \left(\sum_{i=1}^n|x_i|^p\right)^{\frac1p},||x||_∞ = \underset{1≤i≤n}{max}|x_i| $$ Show that $$ \underset{p→∞}{lim} ||x||_p = ||x||_∞$$ ...
0
votes
0answers
28 views

Problems about compactness of sets

1)Let $(X,d)$ be a compact metric space. For each $x$ in $X$, we define $f_x:X\to\mathbb{R}$, $f_x(y)=d(x,y)$. Let $F=\{f_x|x\in X\}$. Prove that $F$ is a compact subspace of $(C(X),d_\infty)$, where ...
1
vote
4answers
58 views

How to exhibit the set of all the limit points of this subset of $\mathbb{R}^k$?

Let $k$ be a positive integer, let $p_0$ be a point in $\mathbb{R}^k$, let $\delta_0$ be a positive real number, and let the set $E$ be defined as follows: $$E \colon= \{ \, p\in\mathbb{R}^k \, ...
0
votes
1answer
32 views

Uniform convergence of a sequence of functions and equicontinuity

Let $(X,d)$ be a compact metric space. I would like to prove that if $(f_n)_{n \in \mathbb{N}}$ is a sequence of continuos functions $f_n:X \to Y$ that converge uniformly in $X$, then $(f_n)_{n \in ...
0
votes
2answers
28 views

Prove that if any closed ball in $X$ is compact then there exists $y\in F$ such that $d(p,y)=d(p,X)$, for $F$ closed, $p$ a point.

Let $(X,d)$ be a metric space, $F$ included in $X$ and closed, p a point in $X-F$. Prove that if any closed ball in X is compact then there exists y in F such that $d(p,y)=d(p,F)$. I have been ...
0
votes
1answer
23 views

Two problems about separability

The statement of the problem is: Prove that a metric space (X,d) is separable iff d is topologically equivalent to d' and (X,d') is totally bounded. (topologically equivalent means they generate the ...
2
votes
1answer
34 views

Understanding an equality with open balls

In order to understand a proof I want to know why the following is true: Let X be a banach space and $x\in X$ with $x\in B(x_0,r)$ (open ball around $x_0$ with radius $r$), then $\frac{r}{2}x-x_0 \in ...
1
vote
3answers
54 views

Is “total boundedness” a topological property? [duplicate]

I have been trying to find a counterexample of ... If $d$ and $e$ are topologically equivalent metrics (they generate the same open sets), and $(X,d)$ is totally bounded then $(X,e)$ is totally ...
0
votes
0answers
45 views

Problems with topologically equivalent metrics

I want to know if the following proof is correct... If (X,d) is separable then, if S is an open cover of X, I can pick a numerable number of open sets in S, such that X is included in their union ...
3
votes
2answers
37 views

Prove if metric space $X$ contains a connected dense subset, then $X$ is connected.

I am attempting to solve the following question. Q. Suppose a metric space $X$ contains a connected dense subset $A$. Show that $X$ is connected. Here is my attempt at a solution: A. Assume ...
0
votes
1answer
26 views

A finite subset of a metric space is closed

I need to prove that a finite set of points $\{a_1,a_2,\dots ,a_n\}$ in a metric space is a closed set. Can it be assumed that $\{a_i\}$ is a closed set in $X$ so that $ X$ would be the finite union ...
1
vote
2answers
19 views

Open sets in Metric spaces

I need to show that if {x} is open in a metric space X for all x in X,then all subsets of X are open in X I am using the definition that a set A is open if ∀a∈A,∃r>0 s.t. Br(a)⊆A. I tried proving ...
0
votes
1answer
45 views

$f:X \rightarrow Y$ is continuous and surjective , X is complete if and only if Y is complete.

Let $(X,\rho)$ and $(Y,\rho')$ be metric spaces. Assume $f:X\rightarrow Y$ is continuous on $X$ and surjective. Are these statements true: If $(X,\rho)$ complete then $(Y,\rho')$ is complete If ...
0
votes
1answer
39 views

Continuity of unique solution to differential equation

Let $f$ be a continuous function on $G$, where $G \subseteq \mathbb{R}^2$ is an open set containing $I \times [a,b]$ where $I:=[x_0-d,x_0+d]$, for some $a,b,d \in \mathbb{R}$ s.t. $a<b$ and ...
0
votes
1answer
17 views

Prove a dieudonne vector space is a metric space

I'm supposed to prove that a dieudonne vector space is a metric space, but I'm stuck on the triangle inequality. I need to show that $d(x,z) \le d(x,y)+d(y,z)$ with $d(x,y)=|x-y|= \sqrt{(y-x|y-x)}$ ...
1
vote
2answers
51 views

How to prove that, the set of all matrices $M_n{\mathbb R}$ with distinct eigen values is dense in $\mathbb R^n$?

How to prove that the set of all matrices $M_n{\mathbb R}$ with distinct eigen values is dense in $\mathbb R^n$? Is there any geometric interpretations behind this.. if it is so, then tell me how to ...
0
votes
1answer
14 views

any complete metric space $S$ can be homeomorphically embedded as a dense subset of a compact metric space $\bar{S}$

How to prove that any complete metric space $S$ can be homeomorphically embedded as a dense subset of a compact metric space $\bar{S}$. I know that a polish space is homeomorphic to a $G_{\delta}$ ...
1
vote
0answers
50 views

How is this topological space different from the euclidean one?

I'm preparing for my topology exam and came across this example which I can't figure out. Let $\mathcal{T}$ be a such family of all sets $U\subset \mathbb{R}^2$ that $U\cap L$ is an open set in L, ...
1
vote
1answer
31 views

Continuity of functionals in function space

I came across this problem and got confused. With the help of folks at MathStackExchange I managed to understand the following: Define $h:C[0,1]\rightarrow \mathbb{R_+}$ by $$h(x)=\sup_{0\leq ...
0
votes
3answers
34 views

continuity of functional in $C[0,1]$

I came across this problem and got confused Problem: Define $h:C[0,1]\rightarrow \mathbb{R_+}$ by $$h(x)=\sup_{0\leq t\leq1}|x(t)|$$Show $h$ is continuous in $C[0,1]$ Attempt: I am a bit confused ...
2
votes
1answer
72 views

Show that $\lbrace 1-\frac{1}{n} \rbrace_n$ does not converge in the Sorgenfry topology.

Consider $\{1-\frac{1}{n}\}_n=\{0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\cdots\}$. If $\{1-\frac{1}{n}\}_n$ converges, then $\{1-\frac{1}{n}\}_n \rightarrow 1$. If it converges, then, by definition, ...
2
votes
0answers
85 views

A question about compact sets: how to prove $g$ must be an isometry

Let $(X,p)$ be a compact metric space. Suppose that $g:X\rightarrow X$ is a function such that for all $x_1,x_2\in X$ we have $p(g(x_1),g(x_2))\geq p(x_1,x_2)$. Prove that, in fact, $g$ must be an ...
0
votes
0answers
9 views

Conditions on the metric function in a flat manifold

It is well known that a manifold is flat iff its Riemann tensor vanishes identically. However, the equation $R^{\mu}_{\nu\rho\sigma}=0$ is a differential equation for the metric tensor $g_{\mu \nu}$. ...
1
vote
1answer
51 views

Existence of CAT(0)-metrics

Given a metrisable, contractible topological space. Under which conditions exists a CAT(0)-metric compatible with the given topology?
0
votes
2answers
26 views

Metric space problem.

Suppose $S$ is a subset of $\mathbb{R}$ and $z$ belongs to $\mathbb{R}$. Then prove that dist $(z, S) \leq |z-\sup S|$ with equality if $z\geq \sup S$. I can easily prove it for $S=\varnothing$. But ...
0
votes
1answer
24 views

Finding connected componets of a set of continuous functions

In the metric space (C[0,1], d∞) consider the set: U= {f in C[0,1]: f(x)≠0 for all x in [0,1]} Prove that U is open and find its connected components. Proving that U is open is easy, but I don't ...
0
votes
3answers
38 views

When talking about a normed vector space, does it's metric always need to be the induced one?

The title basically says it all. If we have a normed vector space, is it possible to work with the space as a metric space with a different metric than the induced one? So if the space is $(X,||\ ...
5
votes
1answer
56 views

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...
1
vote
0answers
25 views

Positive definite functions defined on the embedding of a planar graph in the plane

By way of motivation: Let $f:[0,1]\rightarrow\mathbb{R}^2$ and $g:\mathbb{S}_1\rightarrow\mathbb{R}^2$ be continuous injections, where $\mathbb{S}_1$ is the circle ($\mathbb{R}/\mathbb{Z}$). Then, ...
0
votes
1answer
31 views

The closed and bounded sets are compact in the product topology

Let $X=\mathbb{R}^{\aleph_0}$ with the product topology, it is true that all the closed and bounded (in the uniform sense) sets are compact?
2
votes
3answers
42 views

Show that the sequence of functions $(x_n)_{n≥1}$ in $C[0, 1]$ given by $x_n(t) = t^{2n} − t^{3n} , ∀t ∈ [0, 1]$ is bounded

That is $C[0,1]$ equipped with the supremum metric. I have proven, using derivatives, that each function $x_n$ has a local maximum and local minimum at $(2/3)^{1/n}$ and $0$ respectively. I know ...
1
vote
0answers
41 views

Problem 30 in the Exercises following Chapter 2 in Baby Rudin: How to immitate the proof of Theorem 2.43?

Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following ...
0
votes
1answer
30 views

Contraction Mapping, Metric

Let $X$ = {all continuous functions $f$:[0,1] $\rightarrow$ [0,1]} and let $d$ be the metric on $X$ given by $d$($f$,$g$)= $sup_{t\in[0,1]}$ |$f$($t$)-$g$($t$)| for $f$,$g$ $\in$ $X$ Show that ...
2
votes
2answers
66 views

Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?

Letting $\mathbb{Q}$ be equipped with the Euclidean metric. What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $. Its not ...
0
votes
3answers
50 views

Contraction-like mapping without fixed point?

If $(X,d)$ is a complete metric space and $\xi:\;X\to X$ satisfies: $$d(x,y)<n+1\Rightarrow d(\xi(x),\xi(y))<n$$ $$d(x,y)<1/n\Rightarrow d(\xi(x),\xi(y))<1/(n+1)$$ for all $n= 1,2,\dots$, ...
1
vote
1answer
39 views

Closedness and boundedness in metrizable topological spaces

This is a quick question that I have not managed to answer myself: let $X$ be a metrizable topological space, let $A\subset X$ be a closed, bounded subset. $f:X\to Y$ is a homeomorphism, must $f(A)$ ...
1
vote
0answers
23 views

Structure of the space $X:=\{x \subset \mathbb{R}^3: |x| < \infty\}$, where $|x|$ is the cardinality of $x$

I am interested in the space $$ X:=\{x \subset \mathbb{R}^3: |x| < \infty\}, $$ where $|x|$ is the cardinality of the subset $x$. This is basically configuration space for a quantum system with a ...