(a) Let \(\ell ^1\) and \(\ell ^\infty \) be the Banach spaces of sequences of complex numbers \(\xi =(\xi _1,\xi _2,\ldots )\) such that \[ \xi \in \ell ^1 \quad \text {if and only if}\quad \lVert \xi \rVert _1:=\sum _{i=1}^{\infty }|\xi _i|<\infty \] and \[ \xi \in \ell ^\infty \quad \text {if and only if}\quad \lVert \xi \rVert _\infty :=\sup _i|\xi _i|<\infty . \] Let \((\ell ^\infty )^*\) denote the Banach space of continuous linear functionals, equipped with the operator norm, on \(\ell ^\infty \). Show that the map \(\Lambda :\ell ^1\to (\ell ^\infty )^*\) defined by \[ \Lambda (\xi )(\eta _1,\eta _2,\ldots )=\sum _i \xi _i\eta _i \] is a well-defined norm-preserving injection of \(\ell ^1\) into \((\ell ^\infty )^*\) which is not onto.
Hint: Show that \((\ell ^\infty )^*\) contains nontrivial functionals which vanish on the subspace consisting of those \(\xi \) such that \(\xi _i\to 0\) as \(i\to \infty \).
(b) Let \((X,\mu )\) be a measure space. State precise conditions on \(\mu \), \(p\), and \(q\) which guarantee a norm-preserving isomorphism of \(L^p(X)\) onto \((L^q(X))^*\).