Math 6710 • Applied Linear Operators and Spectral Methods

Let \(X\) be a normed space over \(\mathbb{R}\) and \(Y \subset X\) be a linear subspace. For \(x_0 \in X \setminus Y\) we
define the linear functional

\[f : \operatorname{span}\{Y, x_0\} \to \mathbb{R}\]

by

\[f(y + \lambda x_0) = \lambda, \quad\text{for all } y \in Y, \lambda \in \mathbb{R}.\]

\begin{enumerate}

\item[(i)]
Prove that \(f\) is continuous if and only if \(x_0 \notin \overline{Y}\).
\item[(ii)]
Prove that \(\overline{Y} = \bigcap\{\operatorname{Ker} g \mid g \in X', Y \subset \operatorname{Ker} g\}\), where \(X'\) is the dual of \(X\).


Hint for part \((ii)\): The \(\subset\) direction is relatively easy. For the other direction, note
that as given in part (i), \(f\) satisfies \(f|_Y = 0\) and \(f(x_0) = 1\). The Hahn-Banach theorem
provides an extension of \(f\) to the entire space, use this to prove the other direction by
contradiction.

\end{enumerate}