Duality of l1 and l infinity ($({\ell }^1{)}^{\ast }={\ell }^{\infty }$)

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.

A Banach space has the Schur property when every weakly convergent sequence converges with respect to the norm topology.

A gliding hump argument selects a subsequence and successive finite coordinate blocks so that each selected vector has little mass before and after its assigned block. A bounded dual vector can then align its signs independently on those disjoint blocks.

The l-p sequence space ${\ell }^1$ has the Schur property. If a weakly null sequence stayed bounded below in norm, coordinatewise convergence and summability would select disjoint blocks containing almost all of successive terms. A sequence in ${\ell }^{\infty }$ matching their signs on those blocks would pair uniformly positively with a subsequence, contradicting weak convergence through the duality of l1 and l infinity.