Math 6410 • ODE

\textbf{Distributions.}
Let \(\Omega\subset\R^n\) be an open set.
\begin{enumerate}
\item[(a)] Let \(\{f_n\}\subset C(\Omega)\), with \(f_n\) converging to \(f\in C(\Omega)\)
pointwise, i.e., \(f_n(x)\to f(x)\) for all \(x\in\Omega\). Does it follow that \(f_n\)
converges to \(f\) in the sense of distributions? Prove or give a counterexample.
\item[(b)] Let \(\{\mu_n\}\subset\mathcal{D}'(\Omega)\) with \(\mu_n\) converging to
\(\mu\in\mathcal{D}'(\Omega)\) in the sense of distributions. Let \(\alpha\) be a fixed
multi-index. Does it follow that the derivatives \(D^\alpha\mu_n\) converge to
\(D^\alpha\mu\) in the sense of distributions? Prove or give a counterexample.
\item[(c)] Find a sequence of \(L^1(\R)\) functions which do not converge in \(L^1(\R)\) but
which do converge in the sense of distributions.
\end{enumerate}