Duality of sequence spaces

Coordinate pairing gives $({\ell }^p{)}^{\ast }={\ell }^q$ for $1<p<\infty$, $({\ell }^1{)}^{\ast }={\ell }^{\infty }$, and $(c_0{)}^{\ast }={\ell }^1$.

These are continuous-dual identifications under coordinate pairing, related to but distinct from the definition of an Lp space.

For the Banach space $c$ of convergent complex sequences with the supremum norm, every bounded complex-linear functional has the unique representationRestrict $F$ to $c_0$ and use coordinate duality of sequence spaces to obtain $(a_j)\in {\ell }^1$. Splitting $x$ into its constant limit sequence and a zero-limit remainder forces $a_{\infty }=F(1)-{\sum }_j{a}_j$. For the norm lower bound choose the first $N$ coordinates of $x$ with phases maximizing $a_j{x}_j$ and make its tail the constant phase maximizing $a_{\infty }\lim x$. The finite prefix and limit contributions attain their absolute values; the remaining contribution is bounded by the coefficient tail, which tends to zero. The opposite norm inequality is the triangle inequality. The index at infinity is an independent coordinate of the dual norm, rather than part of the sum of the finite-coordinate coefficients.

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.