Math 6410 • ODE

Let \(\calH\) be a Hilbert space with inner product \((x,y)\) and
let the set \(\{e_n\}_{n=1}^{\infty}\) be orthonormal in \(\calH\). Complete the
following:

\begin{enumerate}
\item[(a)] Prove that the formal series
\[
g=\sum_{n=1}^{\infty}(f,e_n)e_n
\]
converges for each \(f\in\calH\) to an element \(g\in\calH\).

\item[(b)] Let the mapping \(\mathcal{P}\) be defined on \(\calH\) by \(\mathcal{P}f=g\),
\(g=\sum_{n=1}^{\infty}(f,e_n)e_n\). Prove that \(\mathcal{P}\) is an orthogonal
projection on \(\calH\).
\end{enumerate}