0
votes
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 ...
1
vote
1answer
19 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
0
votes
0answers
22 views

Prove the $d_\infty$ metric is finite

I need to show that the $d_\infty$ metric $$d_\infty(x,y) = \sup|x_i-y_i|$$ for all sequences $x$ such that $\sup|x_i| < \infty$ is indeed a metric by checking the metric axioms. I also need to ...
0
votes
0answers
20 views

Prove (X, l2) is indeed a metric space

I am having some trouble proving that little l2 is indeed a metric. I am getting stuck at proving the triangle inequality holds. d2(x,y)= (∑|xi-yi|^2)^0.5 So I started off by "adding zero" |xi-zi| = ...
1
vote
0answers
24 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
0
votes
1answer
44 views

Isometry is not surjective

According to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$ d_{Y}(f(a),f(b)) = d_{X}(a,b) $$ for all $a,b \in X $ I ...
1
vote
2answers
43 views

Strange Property of Ultrametric Spaces and Metric Completion

The following property of ultrametric spaces seems quite strange: (No new values of the metric after completion) Let $x_1, x_2, \ldots$ be a sequence in $X$ converging to $x \in X$. Suppose $a \in ...
1
vote
1answer
31 views

Cauchy sequence of functions and uniform convergence

If $\Omega$ is a bounded domain, and on $C(\bar{\Omega})$ we use the uniform distance $$d(f,g)=\max_{\bar{\Omega}} |f-g|,$$ a Cauchy sequence of functions (w.r.t. the distance $d$) converge and the ...
-1
votes
1answer
44 views

Does there exist a metric under which $\mathbb{R}$ is incomplete? [duplicate]

Does there exist a metric under which $\mathbb{R}$ is incomplete?
1
vote
1answer
37 views

Poincaré inequality on metric spaces

In this book, page 91, example 4.18, it is said that the space $$Y=\{z\in\mathbb{C}:\ |z|=1,\ \arg z\in [-T,T]\}$$ where $3<T<\pi$ does not support a strong Poincare inequality (see page 84 for ...
0
votes
3answers
90 views

modern analysis: metric spaces and $\varepsilon$-neighborhoods

Prove or disprove that $d(f,g) = ({\int_0^1 |f(x)-g(x)|^{2}dx})^{1/2}$, on $C[0,1]$ is a metric. If so, describe the $\varepsilon$-neighborhood.
0
votes
1answer
30 views

Prove $g$ is continous on metric space.

For:$(X,d)$ is a metric space , $f:X\to X$ is a continous function and $g:X\to R, x\mapsto g(x)=d(f(x),x)$. Prove that $g$ is a continous function. Definition: $f$ is continous at $x_0$ if only if ...
0
votes
0answers
22 views

Convergence of difference of two cauchy sequence

Let $(X,d)$ be metric space, not complete, and $x_n , y_n$ be Cauchy Sequences in $X$. Then is $d(x_n,y_n)$ convergent? I know that $d(x_n,y_n) \leq d(x_n,x_m)+d(x_m,y_m) + d(y_m,y_n)$, so it is ...
0
votes
0answers
54 views

sufficient conditions that a function has a fixed point

Let be $(X,d)$ a complete metric space and $f:X\to X$ with $d(f(x),f(y))<d(x,y)$. I want to show that in general $f$ has no fixed point. But if $(X,d)$ is a compact space, indeed $f$ has a ...
2
votes
2answers
51 views

Is this a metric on R?

For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$. Is this a metric on $\mathbb{R}$? It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality ...
2
votes
2answers
31 views

What is this space with infinitely many different points with distance $1$ between any two different points?

I'm reading Mac Lane's: Mathematics, Form and Function: [...] There are also bizarre examples - such as "a space" with infinitely many different points, with distance $1$ between any two different ...
0
votes
0answers
28 views

Sequence Criterion of Continuity and Divergent sequences

A function $f : (X, d_X) \to (Y, d_Y)$ between metric spaces is continuous iff $$ \lim_{n \to \infty} f(x_n) = f(\lim_{n\to \infty} x_n) $$ for each convergent sequence $(x_n)$ in $(X, d_X)$. So if I ...
0
votes
1answer
30 views

Show that $d_g$ is a metric on $l^1$.

On the space $l^1$ of complex valued sequences $(x_n)$ such that $\sum|x_n|<\infty$, define for $x=(x_1,x_2,\cdots)$, $y=(y_1,y_2,\cdots)$ the metric $d_f$ by ...
3
votes
2answers
90 views

Show that $d(x,y)$ in a metric on $X$.

Let $d_a(x,y)=7|x-y|$ and $d_b(x,y)=|x+y|$ be metrics on set $X$. Show that $d(x,y)=d_a(x,y) + d_b(x,y)$ is also a metric on $X$. Would I be correct in writing $d_a(x,y) + d_b(x,y)$ as $7|x-y| + ...
0
votes
0answers
35 views

Example of a sequence function which is unbounded

Let $(X,d)$ be a metric space and $T:X\to X$ be self map on $X$. A map $\phi :X\to[0,\infty)$ is said to be sequence function with respect to $T$ if $$\lim_{n\to\infty}\phi(x_n)<\infty$$ whenever ...
1
vote
1answer
17 views

Metric on half-open interval s.t. subset is open w.r.t. $d$ iff open w.r.t. Euclidean metric

I wish to find a metric $d$ on the space $X = (0,1]$ such that $(X,d)$ is complete and so that a subset of $X$ is open with respect to $d$ if and only if it is open with respect to the Euclidean ...
0
votes
2answers
27 views

Convergence of $f_{n}(x)=\frac{1}{x^2+n^2}$ and $g_{n}(x)=\frac{2nx}{x^2+n^2}$ in sup norm

I need to show that (i) $f_{n}(x)=\frac{1}{x^2+n^2}$ converges to the zero function in sup norm, and (ii) $g_{n}(x)=\frac{2nx}{x^2+n^2}$ does not. Not sure if this is right but would appreciate ...
1
vote
2answers
79 views

Textbook has wrong answer? - Metric spaces “topological properties” (probably trivial for the confident)

In the book there's a table and above it it reads "we have crossed out the wrong answer" meaning the remaining one is right. I dispute this, there are 4, I thought I got the first one right, but I ...
1
vote
0answers
45 views

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
0
votes
1answer
21 views

Showing that a subset of $\Bbb R^3$ defined by an inequality is open

I have a set, which I'm pretty sure is open and here is why I think so. Let $S = ${$(x,y,z) \in \mathbb{R}^3 \colon \frac{e^{x+y^2-z}-1}{x^2+y^2-z^3} > 7$} Now I know that $(0,0,0)$ can't be in ...
0
votes
1answer
32 views

Unable to show a set is open/closed

Im struggling with how to show a set is open or closed.. worse is i have a test on this in a few days.. Here's an example, let $S = [(x,y) \in \mathbb{R}^2 \colon \frac{x}{y} \leq 7 ] $ I have to ...
3
votes
0answers
71 views

Prove that every separable metric space has a countable base.

A collection $\{V_\alpha \}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have $x \in ...
4
votes
0answers
68 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
4
votes
1answer
44 views

$f^{n_i}(x)\to y$ implies $f^{-n_i}(y)\to x$?

Let $(X, d)$ be a compact metric space and $f:X\to X$ be a homeomorphism. If there exists a sequence $n_i$ such that $n_i\to\infty$ as $i\to\infty$ and $x, y\in X$ are such that $f^{n_i}(x)\to y$ as ...
3
votes
1answer
45 views

Analysis question.

Is this set compact? $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$. I know that is closed and bounded so compact but I don't know how to show it is closed and bounded mathematically. This is the graph ...
0
votes
0answers
69 views

Metric-space, counterexample in Arzela-Ascoli Theorem

My book has very few examples, so I would like an example covering this. The theorem is stated as follows. "Let $(X,d_{X})$ be a compact metric space. A subset K of $C(X,\Re^{m})$ is compact if and ...
1
vote
1answer
52 views

Is there a metric space and meanwhile a linear space such that vector addition discontinuous but scalar multiplication operation continuous?

Some special problems about topological groups or topological linear space theory. Recently I have done some study in some respects about topological group, topological linear spaces. And I found it's ...
5
votes
1answer
68 views

Is every $G_\delta$ set the set of continuity points of some function $f$?

I can prove that given a function $f:X \rightarrow Y$, where $X,Y$ are metric spaces, the set $A \subseteq X$ of points on which $f$ is continuous, is $G_{\delta}$. (Take $U_n = \bigcup_{y \in ...
1
vote
1answer
50 views

Is there any non-translation invariant but homogeneous metric linear space?

A metric linear space is a metric space and vector space, and linear operation is continuous regarding to the metric. I know that a homogeneous, translation invariant metric $d$ can be used to define ...
0
votes
1answer
48 views

Energy norm: what is the intuition behind?

Last week I've read the following definition of Energy Norm (along with the definitions of a distance, Euclidean norm, p-norm, etc..: we were talking about metric spaces). Specifically, for $A\in ...
3
votes
1answer
98 views

“Proof” that $g(t) = (t,t,t,…)$ is not continuous with uniform topology

Let $g : \mathbb R \to \mathbb R^{\omega}$ be the function $$ g(t) := (t, t, t, \ldots). $$ If $\mathbb R^{\omega}$ is equipped with the uniform topology, and $\mathbb R$ with the standard topology, ...
0
votes
0answers
42 views

Is an epsilon-net dense in its totally bounded set?

By definition, a totally bounded set A possesses an epsilon-net for every epsilon greater than 0. Does this mean that every point of A is either a limit point of the epsilon-net or a point in the net? ...
0
votes
1answer
58 views

Topology of a subset of continuous functions on the interval $[-1,1]$ on the metric space $(C[0,1],d_ { \infty})$

Problem statement Let $g \in C[-1,1]$. Consider the set $A=\{f \in C[-1,1] : f(x)\leq g(x), \space \forall x \in [-1,1]\}$. $a)$ Prove that on $(C[0,1],d_ {\infty})$, $A^ {\circ}=\{f \in C[-1,1], ...
0
votes
3answers
60 views

Continuous function from real line of the set of all $n \times n$ real matrix.

Let us take usual definitions of continuity on metric spaces and the usual distance metric on $M(n,\mathbb{R})$. I am looking for a continuous mapping $f : \mathbb{R} \rightarrow M(n, \mathbb{R})$, ...
2
votes
1answer
47 views

Prove statement about a sequence of homeomorphisms $f_n:\mathbb R \to \mathbb R$

The problem statement: Let $\{f_n\}_{n \in \mathbb N}$ be a sequence of homeomorphisms from $\mathbb R$ to $\mathbb R$ and let $F$ be a closed subset of $\mathbb R$ that doesn't contain any rational ...
2
votes
1answer
59 views

Proving completeness and compactness of a sequence of metric spaces.

The problem statement Let $(X_n,d_n)_{n \in \mathbb N}$ be a sequence of metric spaces. Consider the product space $X=\prod_{n \in \mathbb N} X_n$ with the distance $d((x_n),(y_n))=\sum_{n \in ...
2
votes
1answer
37 views

Proving two statements about locally compact spaces

The problem statement: Let $(X,d)$ be a locally compact metric space (for every $x \in X$, there exists a compact neighbourhood of $x$) $a)$ Prove that if $K_1 \subset X$ is compact, then, there are ...
1
vote
1answer
58 views

Proving a continuous function $f:K \cup A \to \mathbb R$ is uniformly continuous if $K$ is compact and $A$ is discrete.

Let $(X,d)$ be a metric space. Let $K \subset X$ compact and $A \subset X$: $\exists \delta>0$ such that $d(a,b)>\delta$ for all $a,b \in A$ with $a \neq b$. Consider in $K \cup A$ the induced ...
0
votes
1answer
27 views

Proving the set of “distance functions” on a compact set is a compact set itself

The problem statement. Let $(X,d)$ be a compact metric space and $C(X)=\{\phi: X \to \mathbb R : \phi \text{ is continuous}\}$. For each $x \in X$ we define the function $f_x: X \to \mathbb R$ ...
0
votes
1answer
196 views

Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$

Consider the space $C^1([0,1])$ and the function $d:C([0,1])\times C([0,1]) \to \mathbb R$ defined as $d(f,g)=|f(0)-g(0)|+sup_{x \in [0,1]}|f'(x)-g'(x)|$. Decide whether the metrics $d$ and ...
1
vote
1answer
27 views

Demonstrate that the following metric space is not compact

Let $X$ be a metric space. Show, if there is an $r > 0$ and a sequence $(x_n)$ from $X$ such that $d(x_n,x_m) \geqslant r$ for $n≠m$, then $X$ is not compact. I know that sequentially compact and ...
1
vote
1answer
59 views

What is the best way to define the diameter of the empty subset of a metric space?

This question is related to Why are metric spaces non-empty? . I think that a metric space should allowed to be empty, and many authorities, including Rudin, agree with me. That way, any subset of a ...
2
votes
1answer
35 views

Limit of sums is sum of limits in a metric space

So I'm aware that in a normed space, the limit of the sums is the sum of the limits: For normed space $(X, ||.||)$, if $x_n \rightarrow a$ and $y_n \rightarrow b$, then $(x_n + y_n) \rightarrow ...
0
votes
0answers
122 views

Proving existence of uniformly convergent subsequence of a sequence of functions

Let $\{f_n\}_{n \in \mathbb N}$ a sequence of integrable and uniformly bounded functions $f_n:[a,b] \to \mathbb R$ and for each $n$ let $F_n:[a,b] \to \mathbb R$ such that $F_n(x)=\int_a^x f_n(t)dt$ ...
0
votes
1answer
86 views

Closure of equicontinuous family of bounded functions.

Let $B(x,y)$ be the set of all the bounded functions $f: X \to Y$ ($X,Y$ metric spaces). Prove that if $\mathcal F \subset B(x,y)$ is an equicontinuous family, then $\overline {\mathcal F}$ is ...