Let $X$ be a normed vector space. Define the dual $X^{*}$ of $X$. Define the normed vector spaces $l^{s}=l^{s}(\mathbb{C})$ for all $1 \leqslant s \leqslant \infty$. [You are not required to prove that the norms you have given are indeed norms.]

Now let $1<p, q<\infty$ be such that $p^{-1}+q^{-1}=1$. Show that $\left(l^{q}\right)^{*}$ is isometrically isomorphic to $l^{p}$ as a normed vector space. [You may assume any standard inequalities.]

Show by a similar argument that $\left(l^{1}\right)^{*}$ is isomorphic to $l^{\infty}$. Does your argument also show that $\left(l^{\infty}\right)^{*}$ is isomorphic to $l^{1}$ ? If not, where does it fail?