Write an account of the classical sequence spaces: $\ell_{p}(1 \leqslant p \leqslant \infty)$ and $c_{0}$. You should define them, prove that they are Banach spaces, and discuss their properties, including their dual spaces. Show that $\ell_{\infty}$ is inseparable but that $c_{0}$ and $\ell_{p}$ for $1 \leqslant p<\infty$ are separable.

Prove that, if $T: X \rightarrow Y$ is an isomorphism between two Banach spaces, then

$$T^{*}: Y^{*} \rightarrow X^{*} ; \quad f \mapsto f \circ T$$

is an isomorphism between their duals.

Hence, or otherwise, show that no two of the spaces $c_{0}, \ell_{1}, \ell_{2}, \ell_{\infty}$ are isomorphic.