All Questions
Tagged with metric-spaces sequences-and-series
574
questions
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If $\forall n,\sum_ka_{n,k}^2<\infty$ and $\forall k,a_{n,k}\to b_k$, how to show that $\sum_kb_k^2<\infty$? [closed]
Let $\ell^2$ denote the metric space of all the square-summable sequences of real numbers. Let $p_n = \left( a_{n1}, a_{n2}, a_{n3}, \ldots \right)$ for $n = 1, 2, 3, \ldots$ be a sequence of points ...
- 26.9k
-2
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1
answer
101
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Convergence of $\sum^{\infty }_{n=0} |a_{n}-b_{n}|$ from convergence of $\sum^{\infty }_{n=0} |a_{n}|$ and $\sum^{\infty }_{n=0} |b_{n}|$
Exercise 12.1.15 of Tao's Analysis II book asks the reader to show that the function $d:X\times X\rightarrow \mathbb{R} \cup \left\{ \infty \right\} $ defined by $d\left( a_{n},b_{n}\right) =\sum^{\...
2
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1
answer
55
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Infinite-dimensional cube $[0,1]^\mathbb N$ compact/complete under certain metrics?
Consider the infinite-dimensional cube $[0,1]^\mathbb N := \{ (x_i)_{i=1}^\infty \ | \ 0 \le x_i \le 1 \ \forall i \}.$ This space can be equipped with certain metrics. Below are two of them.
$d_\...
- 1,450
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41
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Find a bounded real sequence, whose range has exactly one accumulation point, but which is not convergent [duplicate]
Example of bounded sequence on $ \mathbb{R}$ s.t. the set $\{X_n : n \in \mathbb{N}\}$ has exactly one accumulation point but $X_n$ is not convergent
My thoughts: Since Xn is bounded it has ...
0
votes
2
answers
92
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A bounded, not convergent sequence in $\mathbb{R}$ whose range has a single accumulation point.
Homework asks me to find a sequence $(x_{\kappa})\in\mathbb{R}$ which is bounded and not convergent, such that, if $A=\{x_{\kappa}:\kappa\in\mathbb{N}\}$ then A has a single accumulation point.
I ...
user1268049
2
votes
2
answers
46
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Homeomorphism from unit ball to itself preserving norm
Problem: Let $B(0,1)\subset (\mathbb R^2, \lvert \cdot\rvert_2)$ be the open unit ball and let $f\colon B(0,1)\to B(0,1)$ be a homeomorphism. Show that if $(x_n)\subset B(0,1)$ is a sequence such that ...
1
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64
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Prove that $(X, d)$ is a complete metric space where $X$ is the set of all real sequences and $d: X \times X \to \mathbb{R}$ defined by...
I am given a metric space $(X, d)$ where $X$ is the set of all real sequences and $d: X \times X \to \mathbb{R}$ the metric defined by
$$
d(x, y) = \begin{cases}(\sup\{n \in \mathbb{N}: x_k = y_k \...
2
votes
1
answer
49
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Understanding a proof that a uniformly Cauchy sequence of continuous functions $\{f_n\}_n$ converges uniformly to a limit function $f$
Suppose $\{f_n\}_n$ is a uniformly Cauchy sequence of continuous function $f_n:\mathbb{R}\to S$ for a complete metric space $S$. I am trying to understand the "standard" proof that the ...
- 1,231
7
votes
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64
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Property weaker than continuity
Let $f:X\to Y$ be a map between matric spaces. I'm interested in the following property:
For every $x\in X$, there is a sequence $x_n\to x$, with $x_n\ne x$ for every $n$, such that $f(x_n)\to f(x)$.
...
- 151
2
votes
1
answer
68
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Characterisation of a proper map
Let $X$ and $Y$ be topological spaces. A continuous map $F:X \rightarrow Y$ is called proper if the preimage of any compact subset in $Y$ is a compact subset of $X$. I wish to understand the ...
- 219
1
vote
1
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77
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Convergence of a sequence w.r.t French Railway Metric
So I'm given the following version of the French Railway Metric:
$$
d^{'}_{f}(z_1, z_2) =
\begin{cases}
|z_1 - z_2| & \text{if } \exists \lambda \in \mathbb{R} \text{ such that } z_1 = \lambda ...
- 456
12
votes
2
answers
709
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Is the metric topology determined by its convergent sequences?
I am aware that in a first countable space (and thus any metric space) is completely determined by its convergent sequences and their limits, i.e.,
If $\tau_1$ and $\tau_2$ are two first countable ...
- 4,119
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1
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63
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Prove that a set of sequences in $l^{2}$ whose sum of squares is bounded by 1 is an open subset of $l^{2}$
Note: Please excuse my poor notation. I only know how to write some symbols in this software, and not very consistently. I hope how I've presented things still makes it clear enough. If you wish to ...
- 275
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44
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What is the metric of the $\mathscr{l}^\infty$ space?
In one exercise, I have to prove that the space $\mathscr{l}^\infty$ (in $\mathbb{R}$) with respect to sup-norm is complete but I'm not conversant with this space. From this Wikipedia's page, I ...
- 607
3
votes
1
answer
115
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Let $X_{n}$ and $Y_{n}$ are two bounded sequence. I need to prove $\sup_{n\in\mathbb N}|X_{n}-Y_{n}|=0$ iff $X_{n}=Y_{n}$ for all $n$
Let $X_{n}$ and $Y_{n}$ are two bounded sequence. I need to prove $\sup_{n\in\mathbb N}|X_{n}-Y_{n}|=0$ iff $X_{n}=Y_{n}$ for all $n$.
I want this proof. Because if you see this in metric space then ...
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