Math 6420 • PDE

Let \(\ell^{1'}\) be the dual space of \(\ell^1\). The objective here is to show that
\(\ell^{1'}\) is isomorphic to \(\ell^\infty\). You may use that a Schauder basis for
\(\ell^1\) (and \(\ell^\infty\)) is \((e_k)\), where \(e_k=(\delta_{kj})\), i.e. the sequence
with zeroes everywhere except for a \(1\) in the \(k\)--th position.
\begin{enumerate}
\item[(a)] Show that any \(f\in \ell^{1'}\) corresponds to a sequence
\(g=(\gamma_k)\in\ell^\infty\).

\item[(b)] Show that any sequence \(g=(\gamma_k)\in\ell^\infty\) corresponds to some
\(f\in\ell^{1'}\).

\item[(c)] Parts (a) and (b) define a bijective linear mapping
\(G:\ell^{1'}\to\ell^\infty\). Prove that \(G\) is an isomorphism by showing
\(\lVert f\rVert_{\ell^{1'}}=\lVert G(f)\rVert_{\ell^\infty}\) for any
\(f\in\ell^{1'}\).
\end{enumerate}