All Questions
Tagged with metric-spaces convergence-divergence
342
questions
2
votes
0
answers
56
views
Is metric incompleteness of $\overline{X}$ always witnessed by some sequence in $X$?
This following conjecture seems intuitively true to me, but I have trouble proving it (and I haven't found it confirmed/disconfirmed in the literature or online):
Let $M$ be an incomplete metric space....
- 357
0
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0
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47
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An increasing sequence of sets must converge? (monotone convergence theorem for any sets)
Let $X$ be a closed convergence compact set. Let $A_1\subset ...\subset A_n\subset ...$ be a sequence of increasing closed subsets in the set $X$.
What other conditions do we need to impose of $X$ in ...
- 828
0
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0
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13
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$\{A_n\}$ is Kuratowski convergent provided whenever hAni hits a basic open set V frequently, then $\{A_n\}$hits V eventually.
we define Kuratowski convergence like this:
Let $ A_1, A_2, A_3,...$ be a sequence of closed sets in a metric space (X,d).
Define the upper and lower closed limits $LsA_n$ and $LiA_n$ of the sequence ...
- 159
6
votes
1
answer
169
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In what metric spaces does bounded + unique limit point imply convergence of a sequence?
A theorem stated by multiple sources is that any sequence of real numbers converges if and only if it is bounded and has a unique limit point. Apostol has a theorem that generalizes half of this: for ...
- 63
1
vote
0
answers
32
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Divergence of gauge kinetic coupling at the AdS boundary
This is the Einstein-Maxwell-Dilaton Gravity action:
\begin{eqnarray*}
S_{EM} = -\frac{1}{16 \pi G_5} \int \mathrm{d^5}x \sqrt{-g} \ [R - \frac{f(\phi)}{4}F_{MN}F^{MN} -\frac{1}{2}D_{M}\phi D^{M}\...
- 121
1
vote
0
answers
31
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convergences of series in two different metrics
for the past few days I've been studying the topic of metric spaces and now making some exercices on it. However I'm struggling with the following exercice: let $l^1(\mathbb{N})$ denote all function $...
- 189
1
vote
1
answer
50
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Convergence of sequence in metric space
Let $C[0,1]$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$. For $n\in\mathbb{N}$ define the following sequence:
\begin{equation}
f_n(t)=\begin{cases}
n^{3/2}t\ \ \ \ ...
- 61
0
votes
1
answer
96
views
Proving a metric space is not complete
Let $X=C[-1,1]$ be the space of real-valued continuous maps on $[-1,1]$, and define \begin{equation}d(f,g)=\int_{-1}^1|f(t)-g(t)|\ dt\end{equation} for all $f,g\in X$. Prove that the sequence $(f_n)_{...
user1022331
3
votes
1
answer
33
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Convergence in $L_1$ with different norms
I saw an example proving that a sequence that converges in $||.||_2$ doesn't necessarily have to converge in $||.||_1$. However, I was wondering if a sequence converges in $||.||_1$, does it converge ...
- 71
0
votes
0
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39
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$f:X\to Y$ is continuous if it maps convergent sequences to convergent sequences.
The Problem: $f:X\to Y$ is continuous if it maps convergent sequences to convergent sequences.
Note the condition is NOT equivalent to the following: if $(x_n)\subset X$ converges to $x\in X$ then $(f(...
- 1,467
1
vote
1
answer
36
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Convergence of a sequence in R implies convergence of another sequence
Let $(\mathbb{R}, \left\| \cdot \right\|_{2})$ be a metric space equipped with the Euclidean metric and:
$$\ a_{n}\to a$$ a sequence converging to a. Then $\forall p\gt 0$ show that:
$$\ p^{a_{n}}\...
- 223
3
votes
0
answers
126
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Do two metrics, which define the same convergent sequences also have the same limit?
I am wondering whether the concept of two metrics defining the "same convergent sequences" (for instance, when we are interested in the metrics being topologically equivalent) also includes ...
- 53
1
vote
1
answer
71
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Doubling measure implies that the measure of any ball $B_{\lambda\delta}$ is bounded by the measure of a smaller ball $B_\delta$ times a factor
Let $(X,d)$ be a metric space and $\mu$ a borel measure in $X$ such that any open ball in $(X,d)$ has a positive and finite measure. Then $\mu$ is called doubling if there exists a positive constant $...
- 861
1
vote
1
answer
42
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Prove that $(\forall n \in \mathbb{N})$ $(\exists k_0 \in \mathbb{N})$ such that $(x_k)_n = (x_{k_0})_n (\forall k \geq k_0) $ in this metric space
Let $X = \{a = (a_n)_{n \in \mathbb{N}} | (\forall n \in \mathbb{N}) a_n\in\{0,1\}\}$ be space with metric $d:X \times X \to \mathbb{R}$ such that
$$d(x,y) = 0$$ when $x = y$ and in other case $$d(x,y)...
- 609
0
votes
1
answer
70
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Discussion of Theorem $3.25$ of baby Rudin's PMA
Theorem $3.25$ of baby Rudin states as follows.
(a) If $\left|a_n\right|\le c_n$ for $n\ge N_0$, where $N_0$ is some fixed integer, and if $\sum c_n$ converges, then $\sum a_n$ converges.
(b) if $a_n\...
- 307