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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If $(X,\| \|)$ be a Normed Linear Space.. Show that if any vector subspace $Y$ of $X$ is open, then $Y=X$

If $(X,\| \|)$ be a Normed Linear Space, Show that if any vector subspace $Y$ of $X$ is open, then $Y=X$ Attempt: Subspace $Y$ of $X$ is open, $=> \exists ~~r >0 ~~\forall~~ y \in Y$ s.t $ ...
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54 views

Hausdorff distance and union of sets

Let $X$ be a metric space; $A_1$, $A_2$, $B_1$, $B_2$ be non-empty subsets in $X$. Let $d(\cdot,\cdot)$ be the Hausdorff distance between sets in $X$. Then $$ d (A_1 \cup A_2 , B_1 \cup B_2) \leq \max ...
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196 views

Converse of a fixed-point theorem

I'm having some trouble furnishing a proof here. Let $(E, d)$ be a metric space such that any $k$-Lipschitz function has a fixed point for $0 < k < 1$. Does it follow, then, that $E$ is ...
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131 views

manhattan to euclidean metric

One may define a graph on a square lattice by taking the nodes of the lattice as graph vertices and the bonds of the lattice as edges. Suppose for simplicity that the nodes have integer $(x,y)$ ...
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52 views

Constructive fixed-point theorems where finite iteration yields the fixed point

I would like to show that $p$, a fixed point of some effective map $f : S\rightarrow S$, can be constructed effectively. Ideally, I would like there to exist a finite $n$ such that $p = f^n(p_0)$, ...
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1answer
35 views

Is mean pairwise distance a metric over subsets of a metric space.

Specifically, I am looking at finite subsets of a set that is a discrete metric space under Jaccard Distance. I'm having trouble proving the triangle inequality or coming up with a counterexample. ...
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3answers
46 views

On Contractive mapping theorem

I am trying to solve one of the problem in the book "Berkeley Problems in Mathematics": Define a sequence of positive numbers as follows. Let $x_0>0$ be any positive number, and let ...
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2answers
36 views

Something not working out for me in the continuity definition

I'm studying analysis and I've ran into this proposition saying that a function from a metric space X to a metric space Y, is continuous if and only if for every open set O in Y, the inverse image of ...
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1answer
39 views

Metric space, inequation

I have a question to the following problem: Let $(X,d)$ be a metrical space and let be $f:[0,\infty)\to[0,\infty)$ twice differentiable with $f(0)=0$, $f(x)>0$ for $x>0$, $f'\geq 0$ and ...
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2answers
47 views

Proving that $X$ is a closed subset if and only if whenever $B_\epsilon (x) \cap X \neq \emptyset $ for every $\epsilon >0$, then $x \in X$.

Suppose $X$ is a subset of a metric space $M$. I would like to prove that $X$ is a closed subset of $M$ if and only if whenever $x$ is a point in $M$ such that $B_\epsilon (x) \cap X \neq \emptyset $ ...
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34 views

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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1answer
99 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
90 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
157 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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27 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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1answer
78 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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36 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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32 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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42 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
76 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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112 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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1answer
59 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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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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50 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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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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1answer
61 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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74 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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25 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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161 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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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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1answer
53 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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1answer
36 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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1answer
188 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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88 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
185 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
54 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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0answers
47 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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1answer
97 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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44 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
27 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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38 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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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
29 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
63 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
157 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
51 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
80 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
42 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
45 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 ...