Math 6710 • Applied Linear Operators and Spectral Methods

Let \(X\) be a Banach space over the complex numbers. Let \(X_{\mathbb{R}}\) be the vector space over
the reals obtained by restricting the multiplication by scalars used to define \(X\) to the
real line. Let \(f : X \to \mathbb{C}\) be a bounded linear functional on \(X\) and let \(\phi : X \to \mathbb{R}\) be its
real part, defined by \(\phi(x) = \operatorname{Re} f(x)\) for all \(x \in X\).
\begin{enumerate}

\item
Prove that \(\phi\) is a linear functional on \(X_{\mathbb{R}}\).


\item
Prove that \(f\) can be reconstructed from \(\phi\) by:

\[f(x) = \phi(x) - i\phi(ix), \quad\text{for } x \in X.\]



\item
Show that \(f\) as a linear functional on \(X\) and \(\phi\) as a linear functional on \(X_{\mathbb{R}}\) have
the same norm, i.e.

\[\|f\|_{X^*} = \|\phi\|_{X_{\mathbb{R}}^*},\]

where \(X^*\) is the dual of \(X\) and \(X_{\mathbb{R}}^*\) is the dual of \(X_{\mathbb{R}}\).
Hint: To prove one of the inequalities, it may be helpful to find \(\lambda \in \mathbb{C}\) such that
\(|f(x)| = f(x/\lambda)\) for a fixed \(x\) with \(f(x) \neq 0\).

\end{enumerate}