Duality of l1 and l infinity
= Duality of l1 and l infinity
{title2=$(\ell^1)^*=\ell^\infty$}
For $y\in\ell^\infty$, the formula $\phi_y(x)=\sum_nx_ny_n$ defines a bounded functional on $\ell^1$ with $\|\phi_y\|=\|y\|_\infty$. Conversely, every $\phi\in(\ell^1)^*$ is represented this way by the bounded sequence $y_n=\phi(e_n)$. This is an <isometric isomorphism of normed spaces>.