Math 6710 • Applied Linear Operators and Spectral Methods

Let \(X\) be a real Banach space and \(X^*\) denote its dual. Let \(C \subset X\) be a convex set
containing the origin. Define

\[
\begin{aligned}
C^* &= \{f \in X^* \mid f(x) \le 1 \text{ for all } x \in C\} \quad\text{and}\\
C^{**} &= \{x \in X \mid f(x) \le 1 \text{ for all } f \in C^*\}.
\end{aligned}
\]


The goal of this problem is to prove that \(C^{**} = \overline{C}\).
\begin{enumerate}

\item
Show that \(C^{**}\) is closed.


\item
Show that \(\overline{C} \subset C^{**}\).


\item
Show that \(\overline{C}\) is convex.


\item
Assume there is \(x_0 \in X\) such that \(x_0 \in C^{**}\) but \(x_0 \notin \overline{C}\). Show there exists an
\(f \in X^*\) such that for all \(x \in \overline{C}\) we have:

\[f(x) < 1 < f(x_0).\]



\item
Explain the contradiction in (d) and how it can be used to conclude with the desired
result.

\end{enumerate}