Math 6420 • PDE

Suppose that \(L\) is a bounded linear operator in a Hilbert space \(H\) with closed range.
\begin{enumerate}
\item[(a)] Prove the Fredholm alternative theorem for solutions of the inhomogeneous equation
\(Lu=f\) with \(u,f\in H\).

\item[(b)] Suppose that \(L\) is invertible and has a complete orthonormal set of eigenfunctions
\(\phi_n\), integer \(n\). Solve the inhomogeneous equation of part (a) using an eigenfunction
expansion.

\item[(c)] Suppose that the adjoint operator \(L^*\) has a nontrivial nullspace
\(\mathcal{N}(L^*)\) spanned by the functions \(\psi_i\), \(i=1,\ldots,m\) and \(f\) does not lie in the
orthogonal complement of \(\mathcal{N}(L^*)\). Let \(L\) have a nullspace spanned by the functions
\(\phi_i\), \(i=1,\ldots,m\). Explain the construction of the smallest least squares solution.
\end{enumerate}