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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is the set of matrices with trace equal equal to zero compact

Is it true that the set of all matrices with trace equal to zero a connected and compact subset of the 2*2 matrices over R?
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1answer
45 views

Metric Space and Open Sets

I'm having trouble figuring out where to go with this problem. Any hints or strategies would be appreciated. I have just the basic definitions for open sets, distance metrics, etc. Consider $\Bbb ...
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1answer
14 views

Continuity of function and its value.

Here's a problem I'm struggling with. Not really sure how to do this. My tools are epsilon delta proofs for continuity and that's about it. Let $f:[0,\infty)\to\Bbb R$ be a function which is ...
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3answers
35 views

New metric spaces from given one

Let (X,d) be a metric space, $f : [0,\infty) \rightarrow [0,\infty)$ continuous differentiable, strict monotone increasingly with $f(0)=0$ and a monotone decreasingly derivative. Prove that $f \circ ...
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1answer
28 views

Translation invariant metrics and topological groups

During a lecture was pointed out that one of the main feature, from a topological perspective, of normed vector spaces is the translational invariance, that is that one can study the topological ...
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1answer
7 views

Hausdorff Distance between Subdifferential sets

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...
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1answer
25 views

Metric Space Proof - Analysis

Let $\mathcal{C}([0, 1])$ be the set of all continuous functions $f : [0, 1] \to \mathbb{R}$. For $f, g \in \mathcal{C}([0, 1])$, define $d(f, g) = \max|f(x) − g(x)|$. Show that $d$ is a metric on ...
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1answer
51 views

Proof of that there is no metric on $\mathbb{R}$ which is equivalent to the natural metric and which induces a metric on $(0,1)$

I want to prove the following statements: For $X:=(0,1),$ prove the following: (a) $d(x,y):=\big|(1/x)-(1/y)\big|$ is a metric on $X.$ (b) The natural metric and $d$ are equivalent. (c) There is ...
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166 views

Homeomorphic metric spaces

I want to examine if $(0,1] $ and $\mathbb R $ are homeomorphic. We work on metric space $(\mathbb R, e)$, where $e$ stands for the euclidean metric. My answer: Let's assume there is a ...
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1answer
32 views

Norm on $\mathbb R^n$ with given unit ball

Consider a finite subset $S$ of $\mathbb Z^n$ such that $-s\in S$ whenever $s\in S$ and $S$ generates $\mathbb Z^n$. What is a norm on $\mathbb R^n$ whose unit ball is precisely the convex hull of ...
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1answer
39 views

Show that it is at most countable

In a space of finite measure, show that a family of disjoint measurable sets with positive measure is at most countable. Could you give me some hints what I am supposed to do??
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1answer
55 views

Continuity of a function?

Let $f: (\mathbb N, e_\mathbb N) \to(\mathbb R,e)$, where $e$ stands for the euclidean metric and $$f(n)=\begin{cases} n,\, n\ge 2\\\\0,\, n=1\end{cases}$$ Is $f$ continuous? Firstly, I can ...
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1answer
40 views

Some Strange Minimum and proof

I read following on Norm chapter in one book. $$\begin{align}\left|\left\|x-y\right\|-\left\|w-z\right\|\right| \leq & \min \{\|x-w\| + \|y-z\| , \|x-z\| + \|y-w\|\}\\\text{ or, }&\min ...
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3answers
38 views

Show that a metric on C[a,b] is given by $d(x,y)=\int_{a}^{b}|x(t)-y(t)|dt$

I am somewhat new to functional analysis (and this site, so please constructively chastise me if I commit any faux pas on here). I am one chapter into Kreyszig (Intro.to Func.Anal.) and I am already ...
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1answer
31 views

Topological Vector Space not induced by Metric

Can anyone give me an example of a Topological Vector Space that is not metrizable? I know that the neighborhood base of $0$ needs to be incountable, and all I can construct then is no topological ...
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1answer
45 views

continuous function from one metric space to another metric space

Is differentiation $f(x) \rightarrow f'(x)$ a continuous function from $C^1[a,b] \rightarrow C[a,b]$ ? Is integration $f(x) \rightarrow \int_a^x \! f(t) \, \mathrm{d}t $ a continuous function from ...
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2answers
55 views

Topologies generated by a metric

Hi I am new to mathematical proofs/notation and am working through John Lee's Introduction to Topilogical Manifolds. This is the question and my attempt. This is not homework. 2.4 Suppose $M$ is a ...
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2answers
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topological properties of a given set

Let us consider the set $X=C[0,1]$ with its sup-norm topology. Let $S $ be the set of all elements $f$ of $X$ such that $\int_0^1 f(t) dt=0$. Is $S $ compact and connected? To show $S$ compact I have ...
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3answers
47 views

Density of sets

I have got a problem on whether a set is dense or not but not quite sure on how to approach it. Consider the space $M_2(R)$ with its usual topology.Consider the set $ S$ of all matrices with both ...
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1answer
36 views

Show that there is a series in R^infinity has some term greater than or equal to 1/n but that also is arbitrary close to the zero sequence.

Consider $\mathbb{R}^\infty = \{(a_n): \sum_{n = 1}^{\infty} a_n^2 < \infty\}$ with the metric $d((a_n$), ($b_n$)) = $[\sum_{n=1}^{\infty} (a_n - b_n)^2]^{1/2}$. Let $A = \{(a_n) : |a_n| < 1/n ...
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Closed or open subsets of $C[a,b]$?

$C[a,b]$ denotes the space of continuous real-valued functions on $[a,b]$. The metric associated with $C[a,b]$ here is $d(f,g)=sup[|f(x)-g(x)|]$ where the supremum is taken over $[a,b]$. $C^1[a,b]$ ...
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1answer
131 views

Why isn't $\,\mathcal C[0,1]$ a Banach space in this unusual norm?

I wish to ask the following question: Let $\mathcal X$ be the normed space $\,\mathcal X=\mathcal C([0,1])$, with norm defined as $$ \|\,f\|= \max_{x\in[0,1]} x^2 \lvert\,f(x)\rvert. $$ Why isn't ...
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1answer
20 views

limit in a metric space

Let $(X, d)$ and $(X, d_1)$ be two metric spaces over the same set $X$. Suppose that a sequence $(a_n)$ in $X$ converges in $(X, d)$ to $l$ and converges in $(X, d_1)$ to $l_1$. Then must $l$ equal ...
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Example of metric space that has same interior and closure as its complement? [closed]

Please provide an example for this : Consider a metric space X. Let S be a subset of X. Then S and S complement both have the same interior and the same closure.
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34 views

In a semi-metric space, need the limit of a sequence be unique as it is in a metric space? Yes

In a metric space (M,d), define what you mean by a bounded set B and by L being the limit of a sequence $\{x_n\} \in M$. What is the sequential definition of a function between two metric spaces being ...
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1answer
43 views

Is the set $E=\{0.a_1a_2… \in \mathbb{R}\mid a_i= 4 \text{ or } a_i=7\}$ dense, compact or perfect?

I want to check my reasoning, I found that it's not dense but it's compact and perfect. $1$- It's not dense for 1 is neither in the set of a limit point of it. $2$- It's compact because it's both ...
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4answers
39 views

Give 3 different examples of semi-metric spaces which are NOT metric spaces.

A semi-metric space (M,d) satisfies all of the conditions of a metric space except it need NOT satisfy $d(f,g)=0 \iff f=g$. Give 3 different examples of semi-metric spaces which are NOT metric ...
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1answer
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Why $\mathbb R$ is not complete with the metric $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$?

Suppose $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$. Prove that $\mathbb R$ is not complete with this metric. This is exercise 12 from chapter 1 from Rudin's Functional Analysis. ...
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What if you use the square root instead of squaring? Claim: $(M,d^\frac{1}{2})$ is not a metric space, if $(M,d)$ is a metric space.

A. Let (M, d) be a metric space and $R>0$ be any real number. Show that $(M, R\cdot d)$ is also a metric space. B. Is $(M,d^2)$ also a metric space if (M, d) is a metric space? If yes, ...
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1answer
30 views

What is the metric on a cone?

I'm trying to learn differential geometry. I thought a cone would be an easy place to start with calculating a metric, shape operator, what have you. First of all, by the way, when I say "cone" I ...
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21 views

Show is a norm and a Banach space

$X=C\left([0,1]\right)$ and $\|\cdot\|: X->\Bbb R$ is a norm defined as $\|f\|= \max\limits_{x\in[0,1]} x^2\|f(x)\|$. I had to prove that is a norm on $X$ but $(X,\|\cdot\|)$ isn't a Banach space. ...
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2answers
30 views

A General Question arises about norms and limits [closed]

Suppose $x_n \to x_0$, i.e. $\lim x_n=x_0$. How we can show that $||x_n|| \to ||x_0||$, that is $\lim_{n \to \infty} ||x_n||=||x_0||$, so $\lim_{n \to \infty} ||x_n|| = ||\lim_{x \to \infty} x_n||$?
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Showing that the following conditions are equivalent: [closed]

Let $(X,d)$ be a metric space. We say that $D \subset X$ is dense in $(X, d)$ if $\bar D = X$. Let $D \subset X$. Show that the following conditions are each equivalent: $D$ is dense in $(X, d)$. ...
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2answers
43 views

Showing this metric space is complete

Let $X=(0,1]$ and $d(x,y)=\left|\frac{1}{x}-\frac{1}{y}\right|$. I've proven $(X,d)$ is a metric space but I don't know how to show its completeness. How can I do that?
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Abstracted Metric and Measure Spaces

As I am just beginning to study general topology and metric spaces in more and more detail, it seems to me that the metric space topology is entirely determined by the properties of $\Bbb R$, since ...
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3answers
72 views

Prove that the distance between 2 Cauchy sequences is convergent.

Here is the exact question: Let $(S,d)$ be a metric space. Let $(p_n)$ and $(q_n)$ be two Cauchy sequences in $(S,d)$(note that these two sequences are not necessarily convergent since $(S,d)$ is ...
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1answer
27 views

Kolmogorov n-width of N+1 dimensional ball

For a normed linear space $\mathscr{X}$, let $\mathscr{A}\subset\mathscr{X}$ and $\mathscr{X}_N$ any $N$-dimensional subspace of $\mathscr{X}$. Define the $n$-width of $ \mathscr{A}$ in $\mathscr{X}$ ...
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2answers
38 views

What does it mean for a metric space to be isometrically embedded in another space?

I understand the definition of an isometric embedding, (an injective, distance preserving map) but I don't understand what it means for a metric space to be isometrically embedded in another space. ...
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Sequences, subsequences, sub-subsequences and convergence.

Take the metric space (X,d). Let $\{{x_n}\}_{n\in\mathbb{N}}$ be a sequence in X. If every subsequence has a sub-subsequence and every sub-subsequence converge to the same $x^*$ prove that the ...
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3answers
104 views

Differentiability of the distance function

Suppose that $d:X \times X \to \mathbb{R}$ is a geodesic distance function on a smooth Riemannian manifold $X$ ($d$ is determined by metric tensor) and $x \in X$ is fixed. What can be said about the ...
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Trigonometric polynomials are dense

Is the set of all trigonometric polynomials in the space of continuous functions on [$-\pi,\pi]$ which are $2\pi$-periodic dense?(with sup-norm topology)Please give hints on how to find a sequence of ...
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1answer
54 views

Metric topology induced by the sum of two metrics

I have to show the following: Let $X$ be a set with metrics $d_1$ and $d_2$ inducing metric topologies $\tau_1$ and $\tau_2$. Define a new metric on $X$ where $d(x,y) = d_1(x,y) + d_2(x,y)$ for ...
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In a metric space with a countable base, how does every open cover has a countable subcover?

Let $X$ be a mertic space, and let $\{V_{\alpha}\}$ be a collection open subsets of $X$ such that, for every $x \in X$ and for every open set $G \subset X$ with $x\in G$, there is some $V_\alpha$ such ...
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1answer
21 views

Construct a bounded set of reals with exactly three limit points

I tried doing that, but I didn't get anything at all. Could you provide me with some hints? What I'm sure of Is that, such a set doesn't contain any interval and it's infinite so I think it's a ...
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0answers
39 views

vector field distance

If we have the vector fields $X=(1,0)$ and $Y=(0, a(x))$ with $a(x)=0$ for $x\lt 0$ and $a(x) =1$ for $x \ge 0$, how can we write explicitly the distance? We have 3 cases: if we have $d(P,Q)$ where ...
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1answer
19 views

Does a Reproducing Kernel Hilbert Space of functions always have a distance defined in it?

Recall that a (reproducing kernel hilbert space) RKHS has two equivalent definitions: 1) Its a Hilbert space of functions $\mathcal{H}$ (i.e. vector space with an inner product $\langle \cdot, \cdot ...
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problem about compactness

If X is compact then so is C[X].(C[X] is the set of all continuous functions over X.) Does there exit a Necessary and Sufficient Condition here ;does compactness of C[X] say anything about X?
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How to measure the similarity or divergence of two distributions with different supports?

Suppose $X$ and $Y$ are two random variables with the distributions $F_X$ and $F_Y$ on the same support $\Theta$.Then KL divergence $D_{KL}(X||Y)$ is a way to measure the statistical distance between ...
2
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1answer
47 views

Closed Ball Complete iff $(M,d)$ is complete

I encountered the following in Carothers' Real Analysis: Prove that $(M,d)$ is complete iff for each $r>0$, the closed ball $B_r=\{y\in M: d(x,y)\leq r\}$ is complete. Attempt/Thoughts: ...
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3answers
38 views

Show $\ell_\infty (M)$ is a Banach Space

I'm working on problems from Carothers' Real Analysis. The following problem is in the section on completions. Given any metric space $(M,d)$, check that $\ell_\infty(M)$ is a Banach space. ...