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
0answers
24 views

Progression of a point along geodesics under the action of hyperbolic Möbius transformations

Suppose that $X$ and $Y$ are two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$ with axes intersecting at, say, the centre $O$ of the hyperbolic disc model. Suppose also that the angle between ...
0
votes
2answers
51 views

Given a metric space $(X,\rho)$, prove that $|\rho(x,z)-\rho(y,u)|\leq{\rho(x,y)+\rho(z,u)}$ for $x, y, z, u\in{X}$.

Obviously it is true, but I'm not sure how to prove it. I'm considering the quadrilateral inequality but so far it has not been helpful. Can anyone give me direction on how to verify ...
6
votes
0answers
237 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha ...
0
votes
1answer
105 views

comparison of 3 topologies on C[0,1]

I have a ring of continuous functions from $[0,1]$ to $\Bbb R$. And two norms $C[0,1]\to\Bbb R$. One is supremum of $|f(x)|,$ the other the value of $\int_0^1|f(x)|$. Then I get a Cartesian product of ...
2
votes
1answer
92 views

Embedding finite (discrete) metric spaces to Eulidean space as isometrically as possible

Let $X = \{1, 2, 3, ..., k\}$ with the discrete metric (distance is 1 for every pair of points). How can this be embedded into $\mathbb{R}^n$ (with the usual metric) such that the embedding would be ...
3
votes
1answer
89 views

I want to prove that the following is a metric space

Let $p \geq 1, a = (a_{1},a_{2}) \in \mathbb{R}^2, b = (b_{1},b_{2}) \in \mathbb{R}^{2}$. Denote $d_{p}(a,b) = (|a_{1} - b_{1}|^{p} + |a_{2} - b_{b}|^p)^{\frac{1}{p}}$. Prove that $(\mathbb{R}^2, ...
2
votes
1answer
173 views

X is a metric space. Y is a closed subset of X such that the distance between any two points in Y is at most 1.

I came across this question in an exam I appeared . The question is as follows :- $X$ is a metric space. $Y$ is a closed subset of $X$ such that the distance between any two points in $Y$ is at most ...
3
votes
0answers
111 views

Can we have an isometric embedding of this metric space into an Hilbert space?

A metric space (from this Q&A), is defined below. I'd like to know if its possible to have an isometric embedding of this metric space into an hilbert space? As per Schoenberg theorem $-d^2(x,y)$ ...
1
vote
1answer
103 views

If the vector space of all real valued continuous functions on the metric space (X,d) is finite dimensional then X is finite set

If $(X,d)$ is a metric space such that $C(X,R)$ is a finite dimensional real vector space, would any one help me to show that $X$ is finite set? $C(X,R)$ denotes the set of all real valued continuous ...
1
vote
1answer
104 views

addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
1
vote
3answers
154 views

Can $\le$ be used insted of < in the definition of continuity?

A common definition of a continuous map $T:M_1\to M_2$ is that for every $x\in M_1$ and every $\epsilon>0$ there exists a $\delta >0$ such that for all $y$ in $M_1$ $$d_1(x,y)<\delta \implies ...
3
votes
2answers
282 views

Any ball is connected?

Let $X$ be a compact , metric space. Assume that the closure of every each open ball it the closed ball with same center and radius. Prove that any ball in this space is connected.
2
votes
1answer
88 views

Superspace as the Hilbert Space for Quantum Gravity

This is a question I've asked in physics.stackexchange, but have obtained no answers: Let $\mathcal{A}$ be the Ashtekar connection. Since $^{(3)}g_{AB}=i\frac{\delta}{\delta\mathcal{A}^{AB}}$ (see R. ...
1
vote
1answer
242 views

Complex plane Riemann Sphere topology

Came across the following statement: Define $B_\infty(a;r)$ be the ball in $C_\infty$ with respect to the metric $d_\infty(z_1,z_2) = \frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2}\sqrt{1+|z_2|^2}}$, show that ...
2
votes
0answers
76 views

Is there a non-complete and non-separate metric space?

Is there a (non-trival) non-complete and non-separate metric space? Some notions are here: math.stackexchange.com/questions/182316.
1
vote
1answer
87 views

characterization of uniquely geodesic normed vector spaces

I am trying to understand the proof of Proposition 1.1.6 in Bridson-Haefliger. They deal with the notion of geodesics in metric spaces, as per the definition here: ...
0
votes
1answer
21 views

Let $a,b \in \mathbb{Z}$ with $a<b$. Determine $ d_n:= | \{ c\in\frac{1}{n}\mathbb{Z} \ | \ a < c < b \} |$

The assignment is: Let $a,b \in \mathbb{Z}$ with $a<b$. For $n\in\mathbb{N}$ determine $ d_n:= | \{ c\in\frac{1}{n}\mathbb{Z} \ | \ a < c < b \} |$. Firstly, I began to see what the ...
0
votes
1answer
84 views

Topology, continuous mapping help!!

Give an example of a continuous mapping $$f:X \to Y; \ \ (X,\ Y\ \text{metric})$$ for which there exists an open subset $U \subset X$ such that $$f(U)=\{\ y \in Y \ \mid \ \exists x \in U: ...
3
votes
0answers
153 views

Metric space $X=(0,1)$ with $d(x,y):= \vert \tan x-\tan y \vert$ complete or not? [duplicate]

Let $X=(0,1)$ be a metric space with metric defined as $d(x,y):= \vert \tan x-\tan y \vert$ for all $x,y \in (0,1)$. The question is whether $X$ is complete with respect to the given metric or not. ...
1
vote
1answer
84 views

If a map on a complete metric space has a contraction property, it has a unique fixed point

I am stuck on the following problem: Prove that if $(X, d)$ is a complete metric space and $f : X\rightarrow X$ is a function with the property that there is a number $A < 1$ such that ...
1
vote
1answer
141 views

Define a subset of a metric space that is both open and closed.

Define a nonempty subset of a metric space that is both open and closed. The real line with the Euclidean metric $d(x,y)=|x-y|$ is open and closed. If you take two real lines, not connected together, ...
1
vote
3answers
95 views

If $f:X\to Y$ is continuous and surjective, and $X$ is metrizable, so is $Y$?

The question is that. I have a continuous map $f:X\to Y$ such that $f(X)=Y$. If $X$ is metrizable, then $Y$ is metrizable? Or at least, can we say that $Y$ is homeomorphic to a metrizable space? I ...
0
votes
0answers
112 views

Hadamard space: property of the Busemann function

I have a question about a property of Busemann functions on Hadamard spaces. Let $X$ be a complete CAT($0$) space. If $r:[0, \infty) \to X$ is a geodesic ray, and $x\in X$ the Busemann function is ...
4
votes
2answers
172 views

Trying to show that $C([0,1])$ is a complete metric space, using the norm $\|f|| = \max_{x\in [0,1]} |f(x)|$.

I think I have this problem almost done. I am taking $C([0,1])$ to be the set of all continuous function $f\colon[0,1] \to \mathbb{R}$. I have already shown that $\displaystyle\|f\| = \max_{x\in ...
0
votes
1answer
35 views

Check whether a sequence belongs to an open ball

How to check if the sequence x=( x1 , x2 ,...) where xn =1-(1/n) belongs to the open ball B(0,1) in the normed space l^∞ of all bounded sequences with the norm defined by ...
4
votes
2answers
717 views

Irrational number and Baire space

How to show that the set of irrational numbers is a Baire space ?
0
votes
1answer
444 views

Unit balls in normed spaces.

Assume we talk about the $n$ dimensional vector space over the reals. It is easy to see that for any norm the unit ball is a convex symmetric set. And here is my question : Let $A$ be a bounded , ...
0
votes
2answers
47 views

Using the limit to prove the equivalence between two metrics

Two metrics $d$ and $p$ in some set $X$ are said to be equivalent if for any sequence $x_k \in X$ the following equivalence holds $$\lim_{k \to \infty}d\left(x_k,x\right)=0 \Longleftrightarrow ...
0
votes
2answers
78 views

Equivalence of two metrics defined on $\mathbb{R}^2$

The two metrics $d_{1}$ and $d_{2}$ are said to be topologically equivalent if they generate the same topology. Suppose $d_1(x,y)=\sqrt{(x_1-y_1 )^2+(x_2-y_2 )^2}$ (euclidean distance) $d_2(x,y) = ...
1
vote
1answer
96 views

Is $\rho(x,y)=(x-y)^2$, with $x,y\in \mathbb{R}^1$, a metric space on $\mathbb{R}^1$?

Obviously it has to satisfy the following: 1) For all $x,y\in X$, $0\le d(x,y)$. (positivity) 2) For all $x,y\in X$, $d(x,y)=d(y,x)$. (symmetry) 3) For all $x,y,z\in X$, $d(x,y)\le d(x,z)+d(z,y)$. ...
2
votes
4answers
81 views

$E$ Closed iff $\partial E \subseteq E$

I'm having trouble verifying my proof, would appreciate some input on this one. Let $(X,d)$ be a metric space with $E\subset X$. Suppose $E$ is closed in $X$, which means that $E=\overline{E}$. By ...
1
vote
2answers
147 views

Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
0
votes
2answers
82 views

$d_1(x,y)=d(x,y)+|f(x)-f(y)|$ is a metric, but what does a “ball” look like?

$d_1(x,y)=d(x,y)+|f(x)-f(y)|$ Suppose $f(x)=1$ if $x\in\mathbb{Q}$ and 0 otherwise for a metric space = $\mathbb{R}$. With this function I am shocked that $d_1$ is a metric. I do not doubt my proof ...
1
vote
1answer
85 views

How is this not a metric?

I would probably get this answer eventually, I pose this question because of the time I have spent looking for why it is not. The metric is: $$d_p(x,y)=(|x_1-y_1|^p+|x_2-y_2|^p)^\frac{1}{p}$$ for ...
1
vote
1answer
75 views

Is this an error by my lecturer? Closure definition

My lecturer defined the closure of a set $M$ to be $\overline{M}=\bigcap \{ F \mid F$ is closed and $F\supseteq M \}$. However, in other modules it has been defined as $\overline{M}=\bigcap\limits_{F ...
3
votes
1answer
89 views

D'Alembertian $\Box$

This question has to do with the D'Alembertian operator on a general manifold with a metric $g_{\mu\nu}$. I understand that the definition of the D'Alembertian is $$\Box \phi\equiv ...
3
votes
2answers
85 views

Prove that there are open set $U$ and $V$ in $X$ such that $x∈U, y∈V$ and $U∩V=∅$

Let $X,D$ be a metric space. Suppose that $x$ and $y$ are two distinct points of $X$. Prove that there are open set $U$ and $V$ in $X$ such that $x∈U, y∈V$ and $U∩V=∅$ My professor gave me a hint ...
1
vote
0answers
101 views

Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
2
votes
2answers
137 views

Show that $d^\ast$ is a metric

For $x$ and $y$ in $R$, let $d(x,y)$ be a metric. Show that $$d^\ast(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ is also a metric. It is fairly straightforward to show that $d^\ast(x,y)=0$ if $x=y$ ...
2
votes
1answer
496 views

Why is the weak* topology not in general metrizable?

A Banach space is a topological group under addition. The dual is a topological group under the weak$^*$ topology. The weak$^*$ topology is weaker than the operator norm topology, so is it ...
1
vote
2answers
87 views

Measure of big discontinuities

Let $D\subset\left[ 0,1\right] $ be a dense set, and $\mu$ Lebesgue measure on $\left[ 0,1\right] .$ Suppose $f:\left[ 0,1\right] \rightarrow\left[ 0,1\right] $ is continuous at each point in ...
1
vote
1answer
162 views

Prove that a set A⊂ℝ is connected iff it is an interval. [duplicate]

I know that how to proof that the interval [a,b] is connected. But how to proof that a set A⊂ℝ is connected iff it is an interval.
0
votes
3answers
62 views

Double Limit implies Successive Limits

I know it seems very stupid question, but is it right that: Suppose $X$ being a complete metric space. Then: $$\lim_{(m,n)}x_{(m,n)}\quad\text{exists} \quad\Rightarrow\quad \lim_n\lim_m ...
2
votes
1answer
101 views

Is there a metric space and meanwhile a linear space such that vector addition discontinuous but scalar multiplication operation continuous?

Some special problems about topological groups or topological linear space theory. Recently I have done some study in some respects about topological group, topological linear spaces. And I found it's ...
2
votes
1answer
50 views

recursive series in a metric-space

Let's say that you have a series in a metrix space, defined recursively with $x_{n+1} = f(x_n)$. Let's also say that you are given the knowledge that this series converges. Is it then possible that f ...
2
votes
1answer
85 views

Cauchyness vs. Double Limits

Maybe there are some textbooks which might treat cauchyness by taking double limits... My question: Is it sufficient and necessary to consider the double limit: $$x_n\quad \text{cauchy}\quad ...
5
votes
1answer
176 views

Is every $G_\delta$ set the set of continuity points of some function $f$?

I can prove that given a function $f:X \rightarrow Y$, where $X,Y$ are metric spaces, the set $A \subseteq X$ of points on which $f$ is continuous, is $G_{\delta}$. (Take $U_n = \bigcup_{y \in ...
1
vote
1answer
191 views

Is there any non-translation invariant but homogeneous metric linear space?

A metric linear space is a metric space and vector space, and linear operation is continuous regarding to the metric. I know that a homogeneous, translation invariant metric $d$ can be used to define ...
1
vote
1answer
54 views

Pairwise Maximum Metric

Well , I had question on vector spaces and maximum metric . Lets us assume a set of vectors of $N$ dimension containing only integers , and let us make a set of vectors then we will calculate the ...
1
vote
1answer
62 views

Convergence of the limits of a double sequence in one variable as a sequence of the other variable

If $a_{n,m}$ is a double sequence in a metric space such that $a_{n,m} \rightarrow_m a_n$ uniformly on $n$ and $a_{n,m} \rightarrow_n a$ for all $m$, then $$a_{n} \rightarrow a.$$ Indeed for any ...