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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Convergence of a function in a metric space to its metric

Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
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105 views

What is the correct distance measure for the (anti) de-Sitter space?

Given these two expressions 1) $\sinh{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1−(t^2−x^2)}}$ 2) $\sin{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1+(t^2−x^2)}}$ for distance $d$ from the origin $(0,0)$ to point $(x,t)$, ...
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Inner product and inequalities

Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
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135 views

What's the relationship between the riemannian metric and Jacobi field?

I encounter to the question in reading the following Excise: Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ be the (geodesic) polar ...
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262 views

the hyperbolic plane is complete

I'd like to prove that the hyperbolic plane is complete in a short, nice way. Here's my proof, but I'm not convinced of its correctness yet. Let $\mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert ...
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79 views

Embedding tree metric isometrically into $\ell_\infty$

I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ...
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210 views

Trouble with some equivalent conditions of compactness

I'm afraid this question may turn out to be a stupid one. Though it is related to a previous question of mine, I'll write it down in full. Let $(X, d)$ be a metric space (MS). I have to prove the ...
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106 views

Probability metric spaces for which $r \leq R$ implies $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$

I'm looking for examples of spaces $X$ such that: $X$ is a probability space. $X$ is a metric space. If $x \in X$ and $0 < r \leq R$ then $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$. I ...
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9 views

non-separable metric space and measurablility of its elements

I'm studying Skorokhod space, which consists of cadlag functions, and I encountered the following sentence: If a metric space $(\mathbb{S}, \mathcal{S}, d)$ is not separable, then functions that ...
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19 views

Physical Meaning of Minkowski Distance when p > 2

Suppose we have two vectors in $u, v \in \mathbb{R}^d$. For $p \geq 1$, the Minkowski distance between these vectors is defined as $ \lVert u - v \rVert_p = \Bigl( \sum_{i=1}^d \lvert u_i - v_i ...
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24 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 ?
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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 ...
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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, ...
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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, ...
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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 ...
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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 ...
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28 views

Prove this map is continuous

$(rcos(t),rsin(t))↦((1/r).cos(t),(1/r).sin(t)), 0≤t≤2pi $ first for $0<r<1$, then for $r>1$ My idea is to say $(rcos(t),rsin(t)) = r .(cos(t),sin(t))$ then the cos and sin map with an ...
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35 views

check proof that $B[a,b]$ is not seperable

This is what I have to prove: prove that metric space $B[a,b]$, $a<b$ is not separable. Where $B[a,b]$ is the set of all bounded and defined functions on $[a,b]$, with the metric ...
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54 views

Calculating Euclidean dissimilarity for a given cluster with itself

Suppose I have clusters $$A= \{(1,1)^T, (1,2)^T\} $$ $$B=\{(2,3)^T, (3,4)^T\} $$ $$C= \{(4,5)^T, (5,6)^T, (1,2)^T\} $$ I wish to use the Euclidean dissimilarity and Average linkage to calculate a ...
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25 views

Continuous functions dense in $L_p(X,\mu)$ if $X$ has a special property

Let $X$ be a metric space endowed with a measure $\mu$ satisfying the following condition: all the open and closed sets of $X$ are measurable and for any measurable set $M\subset X$ and any ...
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37 views

highway metric topologically equivalent to euclidean metric?

Consider the Euclidean metric space $(S, d_1)$ on $\mathbb{R^2}$ and the highway metric space $(S, d_h)$ on $\mathbb{R^2}$, where the highway metric is defined as $$d_h(x,y) = \begin{cases} |x_2 ...
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64 views

Confirm solution to chapter 2, Problem 18 in Rudin's book: principals of mathematical analysis

Is there a non-empty perfect set $E$ in $\mathbb{R}^1$ which contains no rational numbers? My effort: Yes, there is. We take $E_0 \colon = [\sqrt{2},\sqrt{3}]$. Then $E_0$ is non-empty, closed, ...
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22 views

Is the following function uniformly continuous?

I am supposed to prove if the function $ f:X \to \mathbb {R}, f (x) = dist (x, A)$ where $ A$ is an arbitrary subset o the metric space $X $ is uniformly continuous. If both points $ x $ and $ y $ ...
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37 views

Is there any standard procedure to properly define a composite metric?

For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$. So if $x_{1},x_{2}\in A\setminus B$, but they are ...
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52 views

Point-set topology - derived sets, boundaries, etc.

Consider the metric space $(\mathbb R^2,\sigma)$, where $\sigma$ is the discrete metric. Let $$E=\lbrace(x,y)\in \mathbb R^2 : x^2+y^2<1 \rbrace$$ i.e. the unit disc. Find the following sets ...
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38 views

Assumptions of Kolmogorov extension theorem

As far as I know, a class of spaces for which the Kolmogorov theorem works and which is closed under countable products, are the spaces of complete separable metric spaces which are also called Polish ...
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43 views

Reference for convergence properties of the Hausdorff distance

Consider the following properties of the Hausdorff distance in $\mathbb R^n$. Let $\Omega_n \supset \Omega_{n+1} \supset ...$ a sequence of open, convex and bounded sets with ...
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38 views

Metric on total space, connection and Hopf fibration

As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: ...
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Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then ...
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14 views

distance metric between multisets

I am trying to define a distance $F(X,Y)$ between two multisets $X$ and $Y$. For each pair of $x \in X , y \in Y$ there exists a distance function $f(x,y)$ which takes the range of $[0,1]$. An ...
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23 views

Does a strictly convex and weak metrizable unit sphere of a Banach space imply separability?

I want know If $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology. Does a strictly convex and weak metrizable unit ...
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48 views

Inequality proof using the triangle inequality

I am reading Kreyszig's Intro to Functional Analysis and am a bit stoked with one of the problems (problem 12 in section 1.1, page 9): Problem: Given a metric space $(X, d)$, show, using the ...
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56 views

$p$-adic metric proof

I need to prove this, Let $p$ be an odd number. It is defined the function $v_p:\mathbb{Q}\to \mathbb{Z}$ as $$v_p\left(p^n\frac{a}{b}\right)=n, \hbox{ if } \mathrm{mcd}(a,p)=\mathrm{mcd}(b,p)=1.$$ ...
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39 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
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57 views

Gromov-Hausdorff distance between a “Line segment” and a “Zylinder”

I want to prove the following statement: For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has $d_{GH}(X,Y) = \frac{\pi}{2} $ where $d_{GH}$ denotes the Gromov-Hausdorff ...
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27 views

Completeness of Locally Compact Metric Space and Group of Isometries

Let $X$ be a locally compact metric space, and suppose that the group of isometries of $X$ acts transitively. Show that $X$ is complete. (This is 2nd part of a problem. In first part I showed that for ...
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159 views

How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?

I have two segments in $\mathbb{R}^2$ and I would like to define a measure of similarity between the two segments. My idea is that I can apply: a scale transformation $s$ in order to equate the ...
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20 views

How are delta distributions defined in metric spaces?

How are delta distributions defined in metric spaces with continuous metric?
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Has anyone seen this space before? Does it have a name?

See the space below (the set taken as a subspace of the plane). It sort of looks like a comb, but with the wrap-around portion added, and the lower left corner removed. What would be a good name ...
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Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
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33 views

Is this subset of a finite metric space already named?

Given a finite set, $X$, with a metric, $d(x,y)$ defined on it, I am interested in the following subsets: $S_k\subseteq X$ s.t. $\forall x\in X,\exists s\in S_k:d(x,s)\geq k$ Do such constructions ...
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65 views

Proving that there is no norm for the space of real-valued sequences making it a complete metric space.

Suppose I have a vector space $K$ which consists of real-valued sequences with only finitely many non-zero terms. I would like to show that there doesn't exist a norm on $K$ that would make it become ...
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38 views

Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
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42 views

Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open ...
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23 views

Creating a metric from a pseudometric

Given the following definition of a pseudo-metric on the set $X$ : A pseudo-metric on the set $X$ is a map $d:X \times X \to \Bbb R^+$ such that for all $x ,y \text{ and } z \in X :$ (PM1) $x=y ...
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26 views

define distance in a manifold over the reals

G is a Hausdorff manifold over the reals with a finite atlas: $\exists m$ $G=\bigcup_{1 \leq i \leq n}U_i$, $g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}^m$. Can I somehow define a metric inside G, ...
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63 views

Prove that a map is a homeomorphism and the inverse is bounded

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free ...
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36 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
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35 views

Question about finite subcovers

I'm having problems wrapping my head around the part with $\rho_i$.Here goes: $A \subset \mathbb{R}^n$ is compact, $\rho$ is a positive real-valued function defined on $A$. Prove: $\exists$ finitely ...
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42 views

a complete space

Define a set $$X=\left\{f:\mathbb{R}\rightarrow\mathbb{R}|f \mbox{ is n-times continuously differentiable}\right\}$$ equipped with the norm $$||y||=\max_{\begin{subarray}{l} ...