Tagged Questions

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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Sequences in Metric Spaces

Find a metric space $(X,d)$ and a sequence $(x_n)$ in X that has no convergent subsequences but for which the infimum of the set $\{d(x_m,x_m)\mid m$ and $n$ are distinct natural numbers$\}$ is zero. ...
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1answer
11 views

Multiple choice question on continuous function on a unit ball

Pick out true: Let $B$ be the closed unit ball and $D$ be the open unit ball. a.Given a continuous function $g:B\rightarrow \mathbb R$ there always exists a continuous function $f:\mathbb ...
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2answers
28 views

Canonical compactification of a metric space

There are many constructions to produce a compact metric space from an arbitrary metric space (sometimes extra conditions are imposed). But is it possible to compactify a metric space M into M* such ...
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2answers
22 views

Continuity of vector space operations in a normed space

Here's problem 4 immediately following section 2.3 in Erwine Kryszeg's book, Introductory Functional Analysis With Applications: Show that in a normed space $X$, vector addition and scalar ...
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0answers
25 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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2answers
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Prove $d(x,y)=\sup _{n} \left| \sum _{k=1}^{n}(x_k-y_k)\right |$ is a metric

Let $\gamma$ be the set of convergent series.$$\gamma = \{x=(x_k), x_k \in \mathbb{R} : \sum x_k <\infty\}$$ Prove that $(\gamma , d)$ is a metric space, with $$d(x,y)=\sup _{n} \left| \sum ...
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2answers
23 views

Check if the parabola (with an induced topology) $\{(x,y)\in\mathbb{R}^2 | y=x^2\}$ is connected or compact.

i think yes connected but not compact, as it cannot be represented as a disjoint union and there is no finite sub cover. I'm just not sure how to go about proving this i.e. what to actually write ...
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2answers
35 views

Is the map, $ f:(0,1)⊂ \mathbb{R}$ → $(1,∞)⊂ \mathbb{R}$ : $x ↦ 1/x $continuous?

I feel it is, but cannot prove why. Also is it bijective, and is its inverse continuous?
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1answer
24 views

Making an interval with point deleted complete

I am playing around with metric and topological spaces to get a better grasp of them, and I am wondering the following: is it possible to have a metric such that the set $[-1,0)\cup (0,1]$ is complete ...
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1answer
51 views

Show these sets are homeomorphic to eachother

1) {${(x, y) ∈ R^2 |0 < x^2 + y^2 < 1}$} 2) {${(x, y) ∈ R^2 | x^2 + y^2 > 1}$} I've considered mapping r to 1/r, from (0,1) to (1,∞)
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1answer
40 views

Theorem 2.3-2 in _Introductory Functional Analysis With Applications_ by Erwine Kryszeg

Here's the statement of Theorem 2.3-2 in the book mentioned above: Let $(X,||\cdot||)$ be a normed space. Then there is a Banach space $\hat{X}$ and an isometry $A \colon X \to W$, where $W = A(X)$, ...
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3answers
33 views

For $A,B,C\subset X$where $X$ is a metric space under some $d$, check if the triangle inequality holds for $d_m(A,B)=\min_{x\in A,y\in B}\{d(x,y)\} $

$$d_m(A,B)=\min_{x\in A,y\in B}\{d(x,y)\} $$ Is it the case that $$d_m(A,C)\leq d_m(A,B)+d_m(B,C)$$ based on the definition of $d_m$ and the fact that $d$ is already some arbitrary metric on $X$? I ...
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0answers
48 views

Proving symmetry of metric (single linkage between clusters using arbitrary dissimilarity measure)

I am told to assume that our dissimilarity measure $d$ satisfies the properties required of it, what seems to be the definition of a metric: $d(x,y) \geq0 $ and $d(x,y)=0 \Longleftrightarrow x=y$ ...
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1answer
31 views

If $(a_n)$ is Cauchy it has a subsequence $(a_{n_i})$ such that $d(a_{n_{i+1}},a_{n_i})<2^{-i}$ for all $i$.

Let $(X,d)$ be a metric space and $(a_n)$ a Cauchy sequence in $X$. How to show that there exists $n_1<n_2<\cdots$ such that $$d(a_{n_{i+1}},a_{n_i})<2^{-i},$$ for all $i\in\mathbb{N}$? I ...
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1answer
23 views

Show that an irregular 1-set in the plane is totally disconnected

A $1$-set is a Borel set such that $0 < \mathcal{H}^1(A) < \infty$, where $\mathcal{H}^s$ is the Hausdorff measure. Let $A$ be an irregular $1$-set in the plane. Deduce from the theorem below ...
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0answers
34 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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0answers
48 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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1answer
19 views

Properties of metric $d$ in a connected space $X$

If $X$ is a connected metric space, and $p$ is a cut point, such that $X-\{p\} = B \cup E$, and $B,E$ are open disjoint subsets of $X$, can I say that the metric $d$ on $X$ has to make it true that ...
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1answer
39 views

if $X$ has a finite number of isolated points, is $X$ compact?

If every real valued continuous function on $X$ is uniformly continuous is $X$ is compact? Moreover if $X$ has a finite number of isolated points, is $X$ compact now? I think that the answer to the ...
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1answer
36 views

$X $ is complete if every real valued continuous function on $X $ is uniformly continuous

If every real valued continuous function on $X $ is uniformly continuous,then is $X$ complete? My attempt:let $x_n$ be a Cauchy Sequence in $X$. Let $f$ be a real valued continuous function. To show ...
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0answers
17 views

Condition for uniform convergence of Fourier series

Let $f$ be a Lebesgue summable periodic function on $[-T/2,T/2]$. I read in Kolmogorov-Fomin's (p.414 here) that if $f$ is bounded on a set $E\subset[-T/2,T/2]$ and for any $\varepsilon>0$ there is ...
2
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1answer
31 views

Completing the solution, lipschitz maps inducing other maps

Let $(X, d)$ be a metric space, $(E, || \cdot ||)$ a Banach space, $(AE(X), || \cdot ||)$ - as described below. I've already proven that for any Lipschitz function $u: X \rightarrow E $ there exists ...
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1answer
34 views

Continuous and preserves measurability $\implies$ preserves null sets.

Let $X$ be a (Lebesgue-)measurable set of $\mathbb{R}^n$ and $f:X \to \mathbb{R}^n$ continuous function that preserves measurability ($A$ meausurable $\implies f(A)$ measurable). Prove: for all $A ...
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1answer
22 views

Ceiling of a metric is a metric

Suppose $d$ is a metric on the set $X$. For $x$, $y \in X$ define the function $c$ by $c(x,y) = \lceil d(x,y) \rceil$. Show that $c$ is a metric on $X$. (The reason I put the question here is because ...
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2answers
25 views

Unions of closed sets [Topology, S.Willard, exercise 3F]

Can anybody gives me a hint to show that for a family $C_\lambda,\lambda\in\Lambda$ of closed sets in some metric space $X$ such that $d(C_{\lambda_1},C_{\lambda_2})\geq\epsilon$ for all ...
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1answer
43 views

The norm of operator $\mathscr{L}$ on the finite-dimensional vector space $V$ equals the norm of operator restricted by some Invariant subspace.

The norm of linear transformation $\mathscr{L}$ on the finite-dimensional vector space $V$ over $\mathbb{R}$ with standard inner product equals the norm of linear transformation $\mathscr{L}$ ...
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0answers
18 views

Metric and confusion

I define a metric tensor $g_{\mu\nu} = e^{2\Omega(x)}\delta_{\mu\nu}$ Then, I do the following passages that seem to lead to inconsistence and confusion: ...
2
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2answers
18 views

Compactness of two equivalent metric spaces

Let $X$ be a non-empty set. Suppose that $d_1$ and $d_2$ are two possibly different metrics on $X$. Let $\tau_i$ denote the topology generated by the metric $d_i$ ($i\in\{1,2\}$). The following are ...
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0answers
37 views

Empty metric space, complete?

I have two questions. Can the empty set be formed into a metric space? If it exists, is it complete? I have thought that the empty set is a complete metric space, since we can let ...
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5answers
80 views

If $f$ is uniformly continuous on $(a, b)$, then $f$ is bounded on $(a, b)$.

So I know that since f is uniformly continuous on (a, b), then for every $\epsilon > 0$, there exists $\delta > 0$ such that for all x and y in (a, b), if |x - y| < delta, then |f(x) - f(y)| ...
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1answer
40 views

Intersection of metric spaces [on hold]

Do you think that the intersection of two metric spaces (X, $\mathcal T$) and (X, $\mathcal T'$) is a metrizable space, or at least is a Hausdorff space ? If this is not the case, would you have any ...
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2answers
24 views

Continuity in Metric Spaces between two spaces under a function f

Let (X,d) and (Y,e) be metric spaces , and let f:X→Y be a function. Explain but do not prove if the statement is correct. If there exists r>0 so that $e((f(x1),f(x2))$$≤$ $r(d(x1,x2))$for every ...
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1answer
27 views

Projection map is open

Let $X=X_1$ x $X_2$ where $(X_1,d_1)$ and $(X_2,d_2)$ are metric spaces. Equip $X$ with a product metric $d$. Define a map $\Pi_1:X \to X_1$ by $\Pi_1(x_1,x_2) = x_1$. Let $U \subset X$ be open and ...
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1answer
19 views

Continuously differentiable functions dense in $L^2[a,b]$

I read in Kolmgorov-Fomin's Элементы теории функций и функционального анализа (p. 408 here) that the set of continuously differentiable functions are dense everywhere in space $L^1[a,b]$ of Lebesgue ...
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2answers
16 views

Subsets of uniformly discrete sets are closed

Let $(X,d)$ be a metric space. Suppose that $A$ is a countably infinite subset with the property that there exists some $\varepsilon>0$ such that if $a,b\in A$ and $a\neq b$, then ...
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4answers
191 views

Does a continuous point-wise limit imply uniform convergence?

Question Given a sequence of continuous functions $(f_n)_{n \in \mathbb N}$ and define $$ f : X \rightarrow Y, \quad f(x) = \lim_{n \rightarrow \infty} f_n(x) $$ where $X$ and $Y$ are metric spaces. ...
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2answers
42 views

Covering of Complete Metric Spaces

Baire's Theorem says that if $X$ is a complete metric space and $$X=\bigcup_{k=1}^{\infty}A_k,$$ then there exists an $n$ s.t. $\stackrel{\circ}{\overline{A_n}}\neq\emptyset$. However, is it possible ...
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1answer
58 views

Question about the Ascoli-Arzelá Theorem proof

Ascoli-Arzelá Thoerem: Let $K$ be a compact space and $M$ be a metric space and $C(K,M)$ be the set of continuous functions from $K$ to $M$. $H \subset C(K,M) $ is relatively compact if and only if ...
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1answer
39 views

Proof that uniform topology is finer than compact convergence topology.

I've tried a few approaches for the last few hours but nothing really works. I already proved that the compact convergence topology is finer than the pointwise convergence topology, if this helps. To ...
2
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1answer
19 views

Show that the image of a complete metric space under a continuous map is also complete given an additional condition.

This is a problem from revision material for a functional analysis class. Let $(X,d)$ and $(C,p)$ be two metric spaces and let $f:X\rightarrow C$ be a continuous function with $f(X)=C$. Assuming ...
3
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3answers
109 views

Prove that discrete metric space is complete

I understand the proof but I want to confirm one. So in discrete metric space, every Cauchy sequence is constant sequence and that way every Cauchy sequence is convergent sequence. Thus we conclude ...
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1answer
51 views

Prove that $\mathbb{R}- \{x\}$ is not complete. [closed]

Let $x\in \mathbb{R}$. Show that the set $S=\mathbb{R} -\{x\}$ is not complete.
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2answers
46 views

Doubt on Arzela-Ascoli theorem

Consider a sequence of equicontinuous and uniformly bounded functions on a compact set. Under which condition I can say that it has a unique uniformly convergent subsequence ? Or, atleast uniform ...
2
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1answer
23 views

Approximation of $f\in C[a,b]$ by functions constant on intervals of length $(b-a)/2^n$

I read (p. 405 here) that a continuous function on $[a,b]$ can be arbitrarily approximated according to the distance of space $L^2[a,b]$ defined by $d(f,g):=\|f-g\|_2=\int_{[a,b]}|f-g|^2d\mu$ with ...
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2answers
22 views

A metric capped at a maximum value is a metric

Suppose $d$ is a metric on the set $X$ and $R$ is a real number with $R>0$. For $x$, $y \in X$ define the function $d_R$ by: $$ d_R(x,y) = \begin{cases} d(x,y) & \text{if } d(x,y) \leq R \\ R ...
0
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1answer
14 views

Maximum of two metrics is a metric

Let $X$ be a set endowed with two metrics $d_1$ and $d_2$ and for all $x$, $y \in X$ define the function $d(x,y) = \max\{d_1(x,y),d_2(x,y)\}$. Show that $d$ is a metric on $X$. (Note I put up this ...
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1answer
13 views

What is a good simple definition or characterization of 'polyhedral space'?

I would like to give a definition of polyhedral space in $\mathbb{R}^n$ that is easy to understand by people that has some Maths knowledge but are neither expert in Calculus nor any other Mathematics ...
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1answer
45 views

Metric spaces Lipschtiz mapping proof

Prove that the map $f : R^2 → R$ , $f(x, y) = 2 \sin x − y$ is a Lipschitz mapping with Lipschitz-constant $2\sqrt{2}$. You can use the fact that $\sqrt2\sqrt{a^2 + b^2} ≥ |a| + |b|$ So if f(x,y) ...
2
votes
3answers
54 views

An example where $f:X \to X$ is not a contraction map but $f \circ f$ is?

Can anyone give me one example where $X$ is a complete metric space, $f:X \to X$ is not a contraction map, but $f \circ f$ is? I thought in terms of having a unique fixed point, also but couldn't ...