Math 6410 • ODE

Select with a check-mark \checkedchoice{} and solve either B-2.I
or B-2.II:

\choicebox{} \textbf{B-2.I:} Define the Sobolev space \(H^m(\Omega)\) for open
\(\Omega\subset\R^n\). Then:
\begin{enumerate}
\item[(a)] Prove that \(H^m(\Omega)\) is a Hilbert space.
\item[(b)] Give an example of a sequence which shows that the subspace \(C([0,1])\) in
\(L^1(0,1)\) is not complete in the \(L^1\)-norm.
\item[(c)] Compute the distributional derivatives \(\partial f\) and \(\partial^2f\) for
\(f(x)=\lvert x\rvert\) in \(H^2(-\infty,\infty)\). Assume results for the
Heaviside unit step and Delta.
\end{enumerate}

\choicebox{} \textbf{B-2.II:} Define what it means for \(H\) to be a Hilbert space. Then:
\begin{enumerate}
\item[(a)] Explain the meaning of the formula \(H=M\oplus M^\perp\) and give conditions
on \(M\) for when it is true (do not give proofs).
\item[(b)] State the Riesz representation theorem and use (a) to prove it.
\end{enumerate}