 Duality of l1 and l infinity

For $y\in {\ell }^{\infty }$, the formula ${\varphi }_y(x)={\sum }_n{x}_n{y}_n$ defines a bounded functional on ${\ell }^1$ with $\Vert {\varphi }_y\Vert =\Vert y{\Vert }_{\infty }$. Conversely, every $\varphi \in ({\ell }^1{)}^{\ast }$ is represented this way by the bounded sequence $y_n=\varphi (e_n)$. This is an isometric isomorphism of normed spaces.