- #1

Mr Davis 97

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## Homework Statement

Identify ##\bar{c}##, ##\bar{c_0}## and ##\bar{c_{00}}## in the metric spaces ##(\ell^\infty,d_\infty)##.

## Homework Equations

The ##\ell^\infty## sequence space is

$$

\ell^\infty:=\left\{\{x_n\}_{n=1}^\infty\in\mathbb{R}^\mathbb{N}\,:\, \sup(\{|x_n|:n\in\mathbb{N}\})<\infty\right\},

$$

with metric ##d_\infty:\ell^\infty\times\ell^\infty \to [0,\infty)## given by

$$

d_\infty(x,y)=\sup(\{|x_n-y_n|:n\in\mathbb{N}\}),

\qquad\text{ for each $x=\{x_n\}_{n=1}^\infty\in\ell^\infty$ and $y=\{y_n\}_{n=1}^\infty\in\ell^\infty$.}

$$

Set

\begin{align*}

c

&:=

\left\{x=\{x_n\}_{n=1}^\infty\in\ell^\infty:x=\{x_n\}_{n=1}^\infty\text{ converges in $\mathbb{R}$}\right\},\\

c_0

&:=

\left\{x=\{x_n\}_{n=1}^\infty\in c:\,\lim_{n\to\infty}x_n=0\right\},\\

c_{00}

&:=

\left\{x=\{x_n\}_{n=1}^\infty\in c_0\,:\,\text{ there is an $N\in\mathbb{N}$ such that $x_n=0$ for all $n\geq N$}\right\},\\

\ell^\infty_a

&:=

\{x=\{x_n\}_{n=1}^\infty\in\ell^\infty

:|x_n|\le|a_n|,\text{ for each }n\in\mathbb{N}\},

\quad\text{ given }a=\{a_n\}_{n=1}^\infty\in\ell^\infty.

\end{align*}

## The Attempt at a Solution

It's simpler to find the closure of subsets of the real numbers: you just try to find all of elements of a set to which a sequence has a limit. I am having more trouble doing this for spaces which are not subsets of the real numbers. How should I go about doing this? Would the fact that ##\{\text{closure points}\} = \{\text{accumulation points}\} \cup \{\text{isolation points}\}## help? Note that I don't need rigorous proofs; just some intuition that would allow me to suppose what the closures are.

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