Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 21F / Solution

The continuous dual space $X^{\ast }$ consists of bounded complex-linear functionals $\varphi :X\to \mathbb{C}$, with $\Vert \varphi \Vert ={\sup }_{\Vert x\Vert \le 1}|\varphi (x)|$. For $1\le s<\infty$, ${\ell }^s$ consists of sequences with ${\sum }_j|x_j{|}^s<\infty$, norm $({\sum }_j|x_j{|}^s{)}^{1/s}$; ${\ell }^{\infty }$ consists of bounded sequences with supremum norm.

For $a\in {\ell }^p$, define ${\varphi }_a(x)={\sum }_j{a}_j{x}_j$ on ${\ell }^q$. Hölder's inequality gives absolute convergence and $\Vert {\varphi }_a\Vert \le \Vert a{\Vert }_p$. For a finite truncation, choose $x_j=\overline{a_j}|a_j{|}^{p-2}/({\sum }_{j\le N}|a_j{|}^p{)}^{1/q}$ for $j\le N$, zero elsewhere, interpreting the numerator as zero when $a_j=0$. Its $q$-norm is one and its pairing equals $({\sum }_{j\le N}|a_j{|}^p{)}^{1/p}$. Letting $N$ grow proves equality of the norms.

Conversely, for a bounded $\varphi$ let $a_j=\varphi (e_j)$. The same finite test vector gives ${\sum }_{j\le N}|a_j{|}^p\le \Vert \varphi {\Vert }^p$, so $a\in {\ell }^p$. Finite truncations converge in ${\ell }^q$, and continuity gives $\varphi (x)={\sum }_j{a}_j{x}_j$. Thus $\boxed{({\ell }^q{)}^{\ast }\cong {\ell }^p\text{ isometrically}.}$

For ${\ell }^1$, boundedness gives $|a_j|\le \Vert \varphi \Vert$, while any bounded sequence defines an absolutely convergent pairing bounded by $\Vert a{\Vert }_{\infty }\Vert x{\Vert }_1$. Density of finite sequences again proves surjectivity, and single-coordinate tests give norm equality. Hence $\boxed{({\ell }^1{)}^{\ast }\cong {\ell }^{\infty }\text{ isometrically}.}$ The reversed argument fails for ${\ell }^{\infty }$: finite sequences are not dense in its supremum norm; their closure is $c_0$. For example the limit functional on convergent sequences extends by the Hahn-Banach theorem to ${\ell }^{\infty }$, vanishes on every $e_j$, yet takes value one on the constant-one sequence. It cannot be represented by pairing with an ${\ell }^1$ coefficient sequence.