1
vote
1answer
41 views

Does uniform convergence depends on the metric?

Definition: Let $f_n:X\to Y$ be a sequence of functions from a set $X$ to the metric space $Y$. Let $d$ be the metric for $Y$. The sequence $(f_n)$ d-converges uniformly to the function $f:X\to Y$ if ...
0
votes
0answers
14 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
1
vote
1answer
16 views

Net-Complete $\iff$ Sequence-Complete

Prove that a metric space is complete w.r.t. sequences iff it is complete w.r.t. nets! (The converse is trivial of course!)
3
votes
1answer
49 views

How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
2
votes
2answers
67 views

Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?

Often, we use a construction like this: Given a subset $ A $ of a metric space and its limit point $ a $, we know that for every $ \epsilon > 0 $ there is another point $ x $ different from $ a $ ...
0
votes
4answers
80 views

Completeness proof?

First of all, this is not a question about a specific problem, but more about a general technique. When I face a problem such as "show that a metric space $(M,d)$ is complete", the first thing I do is ...
0
votes
1answer
52 views

Show these sequences converge and determine the limit of each.

Using the definition of convergence in metric spaces, show that the following sequences converge and find its limit. 1.) $a_n(x)=\frac{n}{n+1}x^2+\frac{2}{n}x+3$ in $(C[0,1],||.||_1)$ To begin we ...
0
votes
1answer
87 views

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed I know that $A$ and $B$ are closed and bounded, then they are sequentially compact, so $A+B$ also sequentially compact, ...
0
votes
3answers
48 views

Is the convergence of a sequence independent of the chosen metric?

Given a metric $\rho$ on $X$ and a sequence $x_n$ in $X$. Does the convergence of $x_n\to x$ under $\rho$ also imply the convergence to the same limit under any other metric $\sigma$? I don't know th ...
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 ...
3
votes
2answers
70 views

Convergent sequence in product space on $\mathbb{R}^{\omega}$

I am confused about the concept of convergent sequence in product space when learning Munkres's Topology, especially when I am comparing two related exercises of it. The exercise 6 of section 19 ...
-1
votes
1answer
75 views

Describe all the convergent and Cauchy sequences in this metric space

Consider the set of natural numbers $\mathbb N$ with the metric $$d(m,n)=\frac{\left|m-n\right|}{1+\left|m-n\right|}$$ Describe all convergent sequences and all Cauchy sequences in this metric ...
0
votes
1answer
116 views

Convergence in $L^\infty$ norm and continuous function

Let $\mathcal{C}(T)$ be the set of continuous functions on $T$, which is a metric space under the norm $\left\|f\right\|_{\infty}=\sup_{t\in T}\left|f(t)\right|$. Suppose $\{X_{n}\}$ and $X$ take ...
0
votes
1answer
49 views

Countable sum of closed boundary sets

I have to prove that for complete metric space and $f_n$ converge pointwisely to $f$ $f^{-1}(a,b)\setminus Int(f^{-1} (a,b)) $ is countable sum of closed, boundary sets. Here is my solution: ...
2
votes
0answers
81 views

Pointwise convergence on a complete metric space

Let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
2
votes
0answers
76 views

Discontinuous function on Q

let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
0
votes
3answers
49 views

Double Limit implies Successive Limits

I know it seems very stupid question, but is it right that: Suppose $X$ being a complete metric space. Then: $$\lim_{(m,n)}x_{(m,n)}\quad\text{exists} \quad\Rightarrow\quad \lim_n\lim_m ...
2
votes
1answer
45 views

recursive series in a metric-space

Let's say that you have a series in a metrix space, defined recursively with $x_{n+1} = f(x_n)$. Let's also say that you are given the knowledge that this series converges. Is it then possible that f ...
1
vote
1answer
25 views

Convergence of the limits of a double sequence in one variable as a sequence of the other variable

If $a_{n,m}$ is a double sequence in a metric space such that $a_{n,m} \rightarrow_m a_n$ uniformly on $n$ and $a_{n,m} \rightarrow_n a$ for all $m$, then $$a_{n} \rightarrow a.$$ Indeed for any ...
0
votes
1answer
35 views

Algebraic Combinations of Convergent Sequences in Metric Spaces

I'm learning analysis and in going from real sequences to sequences in a general metric space, I noticed that the theorem for the limits of algebraic combinations of convergent sequences was ...
1
vote
0answers
44 views

Uniform implies pointwise convergence

I had a question to show a sequence of functions $(x_n)$ in $C[0,1]$ (equipped with a metric $d$) does not contain a uniformly convergent subsequence. $$ x_n(t) = \ n(1-nt) \ \ \ \ \ \ \forall \ ...
2
votes
2answers
62 views

On eventually constant sequences

It is of course true that in a discrete space a sequence converges iff it's eventually constant. Is the converse true, i.e., if the only convergent sequences in a space are eventually constant, is the ...
0
votes
2answers
59 views

the power series converges in compact convergence topology

Consider the sequence of functions $f_{n}: (-1,1) \rightarrow R$ defined by:$$f_{n}(x) = \sum_{k=1}^{n}{kx^{k}}$$ a) Prove that $(f_{n})$ converges in the topology of compact convergence, ...
5
votes
3answers
159 views

Does $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ imply anything?

Let $(X,d)$ be a complete metric space, $(x_n)_{n\in\mathbb{N}}\subset X$ such that $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ for all $n\in\mathbb{N}$. Since I cannot construct such sequence which is not ...
0
votes
1answer
26 views

Check the convergence of a sequence

Let $X$ =$[0,1]$ and $d(x,y)=|x-y|/(1+|x-y|)$ be the metric defined on $X$. Then check whether the sequence ${x_n = 1/n^2}$ A) Converges in $(X,d)$ B)Does not converge in $(X,d)$ My attempt : I ...
1
vote
1answer
31 views

Closure Criterion for convergence of sequences

I know that $\{z\}=\bigcap\{\operatorname{cl}\,\{x_n\mid n\in S\} \mid S\subseteq \mathbb{N}\ \text{and}\ S\ \text{is infinite}\}$ is one of the criteria's of convergence of sequences in a metric ...
4
votes
1answer
119 views

topology homework

I'm new to topology, and therefore not very good at it yet. I have following questions, that I have ansewer, please help me verify what is not correct and what is missing in my answers. Let $X$ be ...
2
votes
1answer
195 views

Totally bounded subset in complete metric space implies compact?

I am reading the book Elements of Functional analysis by Kolmogorov and Fomin. In chapter 2, section 16 on compact metric spaces the author poses the following theorem which he demonstrates ...
0
votes
1answer
32 views

Convergence criteria for interior

My book gives three equivalent statements as a theorem, under the heading "Convergence criteria for interior" : Suppose X is a metric space, z is in X and S is a non empty subset of X. First,z is in ...
0
votes
3answers
504 views

Why is it that in a discrete metric space only eventually constant sequences are convergent?

I just read this result and was wondering what is the intuitive idea behind this ?
4
votes
2answers
237 views

Bounded derivative implies bounded function?

By the following theorem, it suffices to show that $\{F_n: n\in\mathbb N\}$ is equicontinuous and bounded: If $f_k$ is a sequence in an equicontinuous and pointwise bounded set of maps from a ...
0
votes
1answer
138 views

Is a set of bounded functions bounded?

Please consider the following question (note that $C_b$ is the space of bounded continuous functions): Let $f_k$ be a convergent sequence in $\mathscr C_b(A, \mathbb R^m)$. Prove $\{f_k \mid k = ...
0
votes
1answer
339 views

Convergence of a constant sequence and an eventually constant sequence

I have to prove that a constant sequence and an eventually cinstant sequence is always convergent. I tried to do this as follows : I considered (xn) in X to be a constant sequence such that (xn) = ...
-1
votes
3answers
84 views

Convergence of the sequence, $\frac 1n$

Why does the sequence $\frac 1n$, where $n$ is a natural number , does not converge when R is endowed with the discrete metric ?
1
vote
1answer
89 views

Question on a corollary of the Arzela-Ascoli theorem

I am given a corollary of the Arzela-Ascoli theorem, and I've substantially rephrased it to this: If $S$ is an equicontinuous and pointwise bounded set of functions with domain a compact metric ...
0
votes
1answer
57 views

Closure criterion for the convergence of sequences in a metric space

Suppose $X$ is a metric space, $z\in X$, and $(x_n)$ is a sequence in $X$. Then according to the closure criterion for convergence of $(x_n)$ in $X$ we have that, $$\{z\} = ...
3
votes
1answer
44 views

Is the strong convergence of Borel probability measure metrizable?

In a metric space $(X,e)$, a sequence of Borel probability measure converges strongly, $\mu_i \xrightarrow{s} \mu$, iff for each Borel subset $S \in X$, we have $\lim_{i \to \infty}\mu_i(S) = \mu(S)$. ...
1
vote
1answer
80 views

A question on Cauchy sub-sequences in a metric space $(X,d)$

Let $(X,d)$ be a metric space, and let $(x_n)$ be a sequence in $X$. Prove that if $(x_n)$ has a Cauchy subsequence, then for any decreasing sequence of positive $\epsilon_k \rightarrow 0$, there is a ...
0
votes
1answer
64 views

Another question about showing that a point is not an accumulation point of a given set

Let $C = \{ (\frac 1n , \frac mn) \in \mathbb{R}^2 : m,n \in \mathbb Z , n \neq 0 \} $. I'm trying to argue that each point not on the $y$-axis is not an accumulation point of $C$. Is this ...
6
votes
1answer
92 views

A new kind of convergence

Let $(X,d)$ be a connected metric space. We say a sequence $\{x_n\}_n \subseteq X$ is T-convergent to $x \in X$ if the following is true: $$ \mbox{if} \; a,b \in X \mbox{ and } d(a,x) < d(x,b) ...
7
votes
3answers
265 views

If a fixed point is a limit of a subsequence of iterates, must the whole sequence converge to it?

Say $X$ is a compact metric space, with $f:X \to X$ continuous. Now, for any $x_0$, $f^n(x_0)$ must have a convergent subsequence, say $f^{n_i}(x_0) \to x_\infty$. If we know that any such limit is a ...
2
votes
1answer
64 views

Intuition behind closed subsets of a metric space?

Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space. Consider a metric space $$(X,d)$$ Then consider a subset of this space$$F$$ What the book ...
1
vote
1answer
79 views

Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.

I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
1
vote
2answers
550 views

Continuous functions uniformly convergent to a function, metric spaces, equivalent conditions

Let $X, \ (Y, d)$ be metric spaces, $f_1, f_2, \ldots \ : X \rightarrow Y$ be continuous functions, $f: X \rightarrow Y$ an arbitrary function. Prove that the following condtions are equivalent: 1) ...
0
votes
2answers
67 views

Find a convergent function in metric space

Let $C[−1, 1]$ be the space of continuous functions equipped with the metric $p(f,g) = \max\{|f(x)−g(x)| \mid x \in [−1, 1]\}$. Then the sequence of functions $(f_n):[−1,1]\rightarrow \mathbb{R}$ ...
2
votes
0answers
67 views

convergence in metric space

Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$. Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to ...
2
votes
2answers
84 views

Equality of limits with respect to different metrics.

Suppose that $X$ is a set equipped with two metrics, say $d_1$ and $d_2$. Let $\{x_n\}_{n\in\mathbb{N}}\subset X$ be a sequence of points which converges to $x\in X$ with respect to metric $d_1$. ...
1
vote
2answers
84 views

If $x\in X$ and sequence $(x_n) \in X^{\Bbb N}$ converges in $(X,d)$ , then so does every subsequence of $(x_n)$.

A subequence of a sequence $(x_n)_{n\ge 1}$ is a sequence $ (x_{n_1}, x_{n_2},x_{n_3},...$) where $n_1,n_2,n_3,... \in \Bbb N$ with $n_1\lt n_2\lt n_3\lt ...$ Let $(X,d)$ be a metric space and let ...
1
vote
1answer
34 views

Is a 'normally' convergent sequence still convergent in a metric space which barely excludes its 'normal' limit?

For example, suppose $$ x_n = \frac 1n \\ X = (0, 1)$$ Is $x_n$ convergent in $X$? My guess would be no, since there exists no $x \in X$ which $x_n$ approaches; $x_n$ will eventually surpass any ...
2
votes
1answer
102 views

Correctness of Converging sequence and Adherent Points

$x\in X$ is an adherent point of $A\subset X$ if for every $\epsilon>0$ there exists $y\in A$ s.t. $y\in B(x, \epsilon)$ $B(x, \epsilon)$ is the open ball centered at $x$ with radius $\epsilon$ ...