Math 6420 • PDE

Let \(M\) be a subset of a Hilbert space \(H\). The goal of this problem is to show that:
\[
  \overline{\operatorname{span}M}=H
  \quad\text{if and only if}\quad
  M^\perp=\{0\}.
\]
\begin{enumerate}
\item[(a)] Assume \(\overline{\operatorname{span}M}=H\) and let \(x\in M^\perp\). Show that
\(x=0\).

\item[(b)] Assume \(M^\perp=\{0\}\). Show that \(\overline{\operatorname{span}M}=H\).
\end{enumerate}