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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Proving $d_E$ is a Metric on $\Bbb R^n$

This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1): $d_E(\mathbf {q,p}) = 0 \iff ...
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85 views

Is my answer really wrong?

I posted this answer here which a user pointed out to me is not correct. The question is asking for a proof that a compact metric space is complete. My answer: Note that in metric spaces the ...
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1answer
39 views

How do i show that if every continuous function on $X$ is bounded, then $X$ is compact? [duplicate]

Let $(X,d)$ be a metric space. Assume every continuous function on $X$ is bounded. Prove that $X$ is compact. Well, i don't know which continuous function should i fix to start an ...
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1answer
24 views

Two spaces which are isometric [on hold]

prove that if X and Y are isometric and X is complete,then Y is also complete.Thanks
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21 views

define convergence for a sequence of elements in a metric space [on hold]

Define the convergence for a sequence of real numbers. How will you modify the definition to define convergence for a sequence of elements in a metric space? Give examples of convergent and non ...
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10 views

How can the basis elements for the topology induced by the square metric $\rho(x, y)$ be pictured as square regions in the plane?

$\rho(x, y)=\text{max}\{|x_1 - y_1|, |x_2- y_2|\}$. Suppose $x=(0, 0)$ and you were to 'plot' the basis element $\rho(x, y)<1$, how would you do that?
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0answers
29 views

Hausdorff and Fréchet distances

citation from wikipedia: It is possible for two curves to have small Hausdorff distance but large Fréchet distance Can anybody give me an example where this occurs? (sub-question: is it even ...
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2answers
22 views

Show $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$.

Let $X=(0,\infty)$. Define two metrics on $X$ by $d_{1}(x,y)=|x-y|$ and $d_{2}(x,y)=|x-y|+|\frac{1}{x}-\frac{1}{y}|$ for all $x, y \in{X}$. Let $(x_{n})$ be a sequence in $X$ and $x\in{X}$. ...
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0answers
44 views

On the Gromov-Hausdorff distance

I'm working on my bachelor thesis, and I'm studying principally on two textbooks (Selected Topics on Analysis in Metric Spaces [1] by Luigi Ambrosio and Paolo Tilli and A Course in Metric Geometry [2] ...
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24 views

Rational vs real metric space

How to prevent, in a lesson that deals with basic mathematics, that we give two definitions of a metric ? Because there is one, which takes value in $\mathbb{Q}$, to build $\mathbb{R}$, that we do not ...
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1answer
21 views

Finite set in a metric space [on hold]

Let $X={a,b,c,d}$ and $d$ is trivial. Task Is $A=\{a,b\}$ open or closed? My approach I figured both: since $A$ is an interior to $B(a,1/2)\cup B(b,1/2)$, it is open, and since the complement of ...
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16 views

To show closedness of a subset in a metric spaces

Let $(X, d)$ be a metric space and $p\in X$, $\delta>0$ be fixed. Let $$A=\{q \in X : p \in X, d(p,q)>\delta\}$$ How to show that $A$ is closed? I tried to show that directly by taking $A$'s ...
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40 views

A lemma on function spaces

This is a lemma about function spaces. I'm not really understanding it however. Can someone try explaining it to me? Lemma: let $X$ be in |SET| $(Y, d)$ in |MET|, $f_n$, $f$ is in $Y^X$. Then $f_n\to ...
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1answer
23 views

Is there a distance metric for dot product similarity that preserves the ordering of nearest neighbors?

The dot product and cosine similarity measures on vector space are frequently used in machine learning methods. However, efficient data structures and algorithms often require a metric space distance ...
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1answer
56 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
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16 views

Describe the Unit Ball

I was asked to describe the unit ball in $C(I)$. All I could come up with was that by definition $B_{1}(0):=\{x \in C(I) : ||x||_{\infty}<1 \}$, where $||x||_{\infty}:=\sup_{t \in I} |x(t)|$. Thus, ...
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1answer
26 views

Is the set of all integers with metric $d(m,n)=|m-n|$ a complete space?

Consider the set of integers with a metric defined by $d(m,n)=|m-n|$.Is this set complete with respect to this metric? If it is a metric, then I am stuck here. How can a Cauchy sequence have a limit ...
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20 views

Is there a name for this partial order between metrics?

Suppose we have a set $X$ and two metrics $d_1,d_2$ on it (which may or may not attain $\infty$). Assume furthermore that $d_1,d_2$ have the same metric components (where a metric comoponent is a ...
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1answer
25 views

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact.

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact. Some helpful definitions: bounded - A subset $S$ of a ...
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24 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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1answer
41 views

Quasi-isometry of Schreier coset graphs

Let $G$ be a finitely generated group and $H$ a subgroup. For any choice of a generating set $X$ we can form the Schreier coset graph. Is this graph independent of the generating set up to ...
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52 views

basic question of topology involving compactness and convexity

Consider in $R^n$ a compact and convex set $A$ with $int(A) $ nonempty. then $\overline{int(A)} = A$ ?. i have no idea to prove this. In this direction i only know the following (and hard to ...
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Is a closure a disjoint union of limit points and isolated points

Definition) A point $x\in X$ is a limit point of S if every ball $B(x;r)$ contains infinitely many points from $S$. A point $x\in X$ is called an isolated point of S if $\exists r > 0$ such that ...
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1answer
19 views

What is the metric on ${(L^{p}(\Omega))}^N$? (prove that sobolev spaces uniformly convex)

Here what I've done to prove that Sobolev Spaces $W^{(m,p)}(\Omega)$ are uniformly convex for $1<p<\infty$ Given integers $n\geq 1,k\geq0$, we define $N(n,m)$ as the number of multi-indices ...
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2answers
39 views

Product topology and standard euclidean topology over $\mathbb{R}^n$ are equivalent

I would like to know why the product topology and the standard euclidean topology over $\mathbb{R}^n$ are equivalent. I already found the question here: Showing that the product and metric topology ...
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2answers
886 views

What's the motivation behind metric spaces?

So a metric space is a collection of points together with operations, and where we can determine the distance between any of these points. And it must satisfy 4 axioms which are: For all x in that ...
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11 views

Which Similarity Metric should I use?

I have currently created three similarity metrics, but the point is I have no idea which one I should use and when one is better for use than the other. I am currently using these metrics for people ...
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1answer
40 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
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28 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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167 views

Triangle Inequality on a different normed space

Let $x=(x_1,x_2)$, the norm is given by $[x]=\sqrt{x_1^2+x_1x_2+x_2^2}$ I need to show the triangle inequality holds. So $y=(y_1,y_2)$ and from $[x+y]\le[x]+[y]$ I got ...
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20 views

What does the sup function mean in the context of metrics for probability measures/distances/differences?

I was studying different probability metrics and distances and came across the following source: ...
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1answer
56 views

Using the topology of uniform convergence for functions over non-compact spaces

Let $(X, d)$ be a (complete) metric space, and $C(X)$ be the space of continuous maps over $X$. If $X$ is compact, one often uses the topology of uniform convergence when analyzing $C(X)$. If $X$ is ...
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1answer
52 views

Is the following statement true?

Let $(X_1, d_1),\ldots,(X_n, d_n )$ be metric spaces, $ X: = X_1 \times \cdots\times X_n$ be their Cartesian product with metric $d$. Let $ \pi_i : X \to X_i$ be the projection for $ 1 \le i \le n$. ...
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37 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
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68 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
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1answer
18 views

continuous functions on metric space

Assume $f:X\rightarrow Y$, where $X$ and $Y$ are two metric spaces. If $f(\overline{E})\subset \overline{f(E)}, \, \forall E\subset X$, then how can we prove that $f$ is continuous? Thank you for ...
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1answer
22 views

A complex metric

Given the following definition $d(z , w) = \begin{cases}0 & z=w \\ |z|+ |w| & z\neq w \end{cases}$ I have to prove that $d(z,w)= 0\Rightarrow z = w$ Which is in part of checking that $d$ is ...
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1answer
30 views

A simple question on Hausdorff distance

Let $(A_n)$ be a sequence of compact sets in $R^n$ and consider $K$ and $A$ compact sets in $R^n$. Suppose that $A_n \cup K \rightarrow A \cup K$ in the Hausdorff distance. Then $$ A_n ...
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13 views

Verify global Lipschitz condition

Consider the realized observations $z_1,...z_i,...,z_n$ i.i.d. Let $$ \hat{Q}_n(\theta)=\frac{1}{n} \sum_{i=1}^{n} 1\{z_i\in S(\theta)\} $$ I have to verify the following global Lipschitz condition: ...
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1answer
27 views

The Open Set $X-\lbrace x \rbrace$

I am task with proving the following: if $x \in X$ then $X- \lbrace x \rbrace $ is an open set I kind of have an idea but I am unsure about it and how to express it. I was thinking about using the ...
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1answer
41 views

Using Cantor's intersection theorem

Assume $f: X \rightarrow X$ is a continuous map where X is a compact metric space. Prove that there exists a non-empty set $A \subset X$ such that $f(A) = A$. (Hint: Set $F_1 = f(X), F_{n+1} = ...
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1answer
35 views

About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
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1answer
34 views

How to show that $(C^0((a,b)), d_\infty)$ is not a metric space

Let $d_\infty:C^0([a,b]) \times C^0([a,b]) \to [0,\infty)$ be defined as $$ d_\infty(f,g)=\sup\limits_{x \in [a,b]} \left\{ |f(x) - g(x)| \right\} $$ I have already shown that $(C^0([a,b]), ...
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1answer
30 views

Show that the interior of the set A is empty?

Consider $A = \{(x, \sin\frac{1}{x}) \mid 0< x \leq 1 \}$, a subset of $\mathbb R^2$. Find int($A$). We can see graphically that the interior of $A$ is definitely empty, but I want to check by the ...
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1answer
28 views

Prove that metric space is complete

I have metric space: $$ X = <[0,+\infty), \rho>, \rho(x,y) = |ln(1+x) - ln(1+y)|$$ I know it is complete, but I don't know how to prove it. How can I prove that fact?
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1answer
44 views

Set theory: not really understanding what the question is asking..

Here is the problem. Let $M$ be the metric space of all real numbers, and let $x_0 \in M$. Prove that there exist exactly two isometries of $M$ that leave $x_0$ fixed. I am having trouble ...
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1answer
37 views

Proving the distance function $|f - g|_u = \sup \{ |f(x) - g(x)|: x \in S \}$ defines a metric space

Let $S$ be a closed and bounded subset of $\mathbb{R}$. Define the "functional distance" between $f$ and $g$, both functions from $S$ to $\mathbb{R}$, to be $$ |f - g|_u = \sup \{ |f(x) - g(x)|: x ...
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2answers
23 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
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1answer
30 views

Are these metrics complete?

Determine if these subsets of R are complete with the Euclidean metric? a) $[0,\infty)$ b) $(0,\infty)$ I know the definitions of completeness and I know the Euclidean metric, but don't know how to ...
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1answer
31 views

Prove that if $B(x,r)$ and $B(x',r')$ are disjoint $\Longleftrightarrow d(x,x') \ge r+r'$

Assuming that $d(x,x') \ge r+r'$, and proving that they are disjoint is easy. It's the other side that I'm having difficulty with. This seems like a really easy problem, but i'm having difficulty ...