For a sequence $x=\left(x_{1}, x_{2}, \ldots\right)$ with $x_{j} \in \mathbb{C}$ for all $j \geqslant 1$, let

$$\|x\|_{\infty}:=\sup _{j \geqslant 1}\left|x_{j}\right|$$

and $\ell^{\infty}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C}\right.$ for all $j \geqslant 1$ and $\left.\|x\|_{\infty}<\infty\right\}$.

a) Prove that $\ell^{\infty}$ is a Banach space.

b) Define

$$c_{0}=\left\{x=\left(x_{1}, x_{2}, \ldots\right) \in \ell^{\infty}: \lim _{j \rightarrow \infty} x_{j}=0\right\}$$

and

$$\ell^{1}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C} \text { for all } j \in \mathbb{N} \text { and }\|x\|_{1}=\sum_{\ell=1}^{\infty}\left|x_{\ell}\right|<\infty\right\}$$

Show that $c_{0}$ is a closed subspace of $\ell^{\infty}$. Show that $c_{0}^{*} \simeq \ell^{1}$.

[Hint: find an isometric isomorphism from $\ell^{1}$ to $\left.c_{0}^{*} .\right]$

c) Let

$$c_{00}=\left\{x=\left(x_{1}, x_{2}, \ldots\right) \in \ell^{\infty}: x_{j}=0 \text { for all } j \text { large enough }\right\} .$$

Is $c_{00}$ a closed subspace of $\ell^{\infty} ?$ If not, what is the closure of $c_{00} ?$