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.

learn more… | top users | synonyms (1)

2
votes
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 ...
0
votes
1answer
44 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 ...
2
votes
2answers
54 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 ...
2
votes
2answers
42 views

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 ...
2
votes
3answers
46 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 ...
1
vote
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 ...
4
votes
0answers
45 views

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 $[a,b]$. $C^1[a,b]$ ...
1
vote
1answer
118 views

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

I need to answer the following question: Let $X$ be the normed space $X=C([0,1])$, with norm defined as $$ \|\,f\|= \max_{x\in[0,1]} x^2 \lvert\,f(x)\rvert. $$ Why isn't this a Banach space?
2
votes
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 ...
-1
votes
3answers
34 views

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.
0
votes
2answers
33 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 ...
1
vote
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 ...
1
vote
4answers
38 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 ...
0
votes
1answer
47 views

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. ...
0
votes
1answer
20 views

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, ...
1
vote
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 ...
0
votes
0answers
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. ...
3
votes
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 $\lim_{n \to \infty} ||x_n|| = ||\lim_{x \to \infty} x_n||$?
0
votes
2answers
46 views

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 $(X, d)$. ...
1
vote
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?
1
vote
2answers
79 views

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 ...
2
votes
3answers
69 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 ...
1
vote
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 $\mathscr{X}$ ...
0
votes
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. ...
0
votes
0answers
39 views

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 ...
3
votes
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 ...
0
votes
0answers
17 views

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 ...
1
vote
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 ...
0
votes
2answers
32 views

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 ...
0
votes
1answer
20 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 ...
0
votes
0answers
38 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 ...
1
vote
1answer
17 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 ...
0
votes
0answers
19 views

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?
0
votes
0answers
17 views

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
votes
1answer
46 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: ...
1
vote
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. ...
0
votes
1answer
28 views

Prove that S is closed IFF S contains all of its limit points

I've seen the solution of this problem involving the closure of S; however, I solved it differently the first time through. Is this a valid approach? Proof: Assume S is closed, then $S^{c}$ must be ...
1
vote
2answers
31 views

Which properties must a function, $f$, fulfill for $d$ to be a metric

I have two questions I need to answer: Let $\mathbb{X}$ be a set and $f : \mathbb{X} \rightarrow \mathbb{R}$ a function. Define $$ d : \mathbb{X} \times \mathbb{X} \rightarrow \mathbb{R}, ~d(x,y) ...
0
votes
1answer
28 views

Metric spaces, Heine-Cantor and boundness

Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces and let $(D,d_X|D)$ be a metric subspace of $(X,d_X)$. Consider a function $f: D \to Y.$ If D is compact and f is continuous, then f is uniformly ...
4
votes
1answer
160 views

Completeness of the space of sets with distance defined by the measure of symmetric difference

Let $m$ be the measure defined on the set semiring $\mathfrak{S}_m$ and $m'$ its extension to the minimal ring $\mathfrak{R}(\mathfrak{S}_m)$. I read that $m'(A\triangle B)$ can be used as a distance ...
0
votes
2answers
61 views

product of closed sets

If A and B are two closed sets of $R$ is A.B closed? By A.B I mean the set $\sum_{i=1}{^ n} a_ib_i$ where $a_i \in A,b_i\in B,n\in N$ How to view A.B geometrically? I am new to this subject.Sorry ...
3
votes
1answer
20 views

Cauchy Sequences Lemma in Vector Space E

I ran into a Lemma. Suppose $||.||_1$ and $||.||_2$ are two norms in vector spapce E, such that $||.||_1$ and $||.||_2$ are equivalent norms and {$x_n$} is an equivalent in E, then {$x_n$} is ...
4
votes
3answers
32 views

Cauchy Sequence some challenge

i read this sentence in one of math books: ‌Every convergent sequence in metric space is a cauchy sequence. would you please some one add more detail, why this is true? thanks.
-1
votes
1answer
26 views

Sequences of functions and convegence

Let $(X,d_X)$ and $(Y,d_Y)$ be two metric space, and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$. Let $\left \{ f_n \right \}$ be a sequence in $C(X,Y)$that converges to some $f^0 ...
0
votes
1answer
30 views

$\Bbb{Q}$ is not complete: Carification regarding a proof

In class today we proved that $\Bbb{Q}$ is not complet, you used the fact that $$ \sum_{k=0}^N\frac{1}{k!}\underset{N\to+\infty}{\longrightarrow}e\notin\Bbb{Q}.$$ After that I was perplex to prove ...
4
votes
1answer
29 views

Prove some Equivalences Norm

Suppose $X=R^2$ and $x=(x_1, x_2)$. I see the following are equal EDIT: ( equivalence). why? i couldent find any proof to satisfy me. any hint or idea or proof highly appreciated. $||x||_1= |x_1| + ...
2
votes
3answers
59 views

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is continuous , is $f(x_n)$ a cauchy sequence?

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is a continuous function where $T$ is an another metric space , is $f(x_n)$ a cauchy sequence? ...
0
votes
1answer
35 views

Every infinite subset of E in R having a limit point in E implies E is closed and bounded

Every infinite subset of E in R having a limit point in E implies E is closed and bounded. Could you please help with a formal proof of this result ?
0
votes
1answer
21 views

Unit Ball in 1 norm is open in ($C[0,1] , || \quad ||_{\infty}$)

Claim $B_1(0,1) := \{ f \in C[0,1] ; ||f||_{1} < 1 \} $ in $(C[0,1],||\quad || _1)$ is open in $(C[0,1],||\quad || _{\infty}).$ We need to take any $f \in B_1(0,1),$ and we have to find an ...
0
votes
1answer
44 views

Why are $\{0\}$ and $\{1\}$ open subsets of the discrete metric space $\{0,1\}$?

$\{0\}$ and $\{1\}$ open subsets of the discrete metric space $\{0,1\}$. Let $S$ be a metric space. Then a subset $A \subseteq S$ is considered open if $\forall x \in A, \exists r>0$ such that ...