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Consider a successions space $l_p$, for $p\geqslant1$ in which $x=(x_1,x_2,...,x_k...)$ and $\lim_{n\to\infty} x_n=x_0$. The metric defined is the following $\rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_k-y_k|}{1+|x_k-y_k|}$ Since the series converges $\forall \epsilon>0\exists\:n_0,\forall n\geqslant n_0$ $\rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_{n,k}-x_{0,k}|}{1+|x_{n,k}-x_{0,k}|}<\epsilon$.$k=1,2,3...$

Question:

I do not understand the $k$ in the following metric $\rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_{n,k}-x_{0,k}|}{1+|x_{n,k}-x_{0,k}|}$. Does the $k$ indicator implies the sequence (series) is defined on $\mathbb{R}^n$? Does the $k$ stands for each variable in which the sequence converges in the following way $|x_{n,k}-x_{0,k}|$ for each $k$? If not what is the meaning of $k$?

Pedro Gomes
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  • $x_{k}$ is the $k^{\text{th}}$ element of the sequence $x$. Similarly for $y$... – Mark Feb 15 '18 at 18:43
  • @Mark Why does the author write $x_{n,k}$ and $x_{0,k}$ if $n$ and $0$ must equal some element of $k$? – Pedro Gomes Feb 15 '18 at 19:00
  • @PedroGomes Can you refer to your source? – user144410 Feb 18 '18 at 16:16
  • What does $x_{n,k}$ mean? It seems as if you've only defined the sequence ${x_k}$. – David Feb 18 '18 at 16:18
  • @user144410 These are notes from a class of Introduction to Functional Analysis. – Pedro Gomes Feb 18 '18 at 16:21
  • @PedroGames, If possible you should share it here; otherwise, you need to clarify your question: what is the presented argument about? Is it showing that the metric is well-defined? It seems to me that you are talking about subsequences here when you use the notations $x_{n,k}$? – user144410 Feb 18 '18 at 16:24
  • @David Thanks for your reply. But $x_{n,k}$ meaning is precisely what I want to know. I began to think it was a sequence defined in a $\mathbb{R}^m::,m\in\mathbb{N}$. so the author was using $k$ to include more then $m$ dimensions. But that sounds ,I guess, a little bit far-fetched. – Pedro Gomes Feb 18 '18 at 16:24
  • @user144410 The author introduces the metric in the following way:$x=(x_1,x_2,...,x_k,...)$ a sequence $\lim_{n\to\infty}x_n=x_0$, then $x_n=(x_{n,1}...,x_{n,k},...)$ then the author shows that as $|x_n(k)-x_0(k)|\to 0::N\to \infty$ so does $ \rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_{n,k}-x_{0,k}|}{1+|x_{n,k}-x_{0,k}|}<\epsilon$. I guess that the sub-sequence idea that you proposed might be right. – Pedro Gomes Feb 18 '18 at 16:35
  • @PedroGomes Alright. I think its an argument about the completeness of the space under the given metric. See the answer that I will refer to below. – user144410 Feb 18 '18 at 16:37

2 Answers2

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An answer to your questions is given by the accepted answer of the following question: The space of sequences as a complete metric space

You need to notice that there are two notion of convergence here: pointwise convergence, and the convergence of the sequence itself as an element of the $\ell_p$ space. That is: \begin{equation} x_n := \{x_{n,k}\}_{k=1}^\infty \in \ell_p \end{equation} $\{x_n\}$ is a sequence of sequences: $x_1$ is the first sequence, $x_2$ is the second ..etc each of them has infinite coordinates since they are elements of the space $\ell_p$.

user144410
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  • Thanks for the answer. How could not I see it? Finally I understood as in the function space, it is the distance between the function, in $l_p$ is the distance between the sequences. Tomorrow I award you the bounty. Thanks! – Pedro Gomes Feb 18 '18 at 18:21
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Really I want us to look into it this way. Let

$ \textbf{x}=(x_{1},x_{2},...,x_{n}, ...)$ be a vector in an infinite dimension. The sequence of vector $ \textbf{x}$ can be denoted by $$ \{ \textbf{x}_k \} = \{ x_{n,k}\} = \{ x_{0,k},x_{1,k},...,x_{n,k},...\} $$ so, in the metric in question, $n$ is just a subscript denoting the $n^{th}$ component of the vector $\textbf{x}$. And $k$ is a subscript that denote the $k^{th}$ term in the sequence of vector $\textbf{x}_k$. In fact, $ |x_{n,k}-x_{0,k}|$ can be seen as sequence of distances between each term $x_{k,n}$ and a particular point $x_{0,k}$

Tunde Ishola
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