(1) Let $u_n$, $u \in C^2([0,1])$ and $u_n \to u$ in $L^2([0,1])$. If $u_n'' \to v$ in $L^2([0,1])$, then could we say $v = u''$?
(2) Also, could we choose $u_n \in C^2([0,1])$ such that $u_n \to u$ in $L^2([0,1])$ and $u_n'' \to v$, where $v$ is not continuous.
These questions are from John K. Hunter's Applied Analysis. Any help would be really appreciated!
For (2) take $u_n(x)=\frac {x^{n}} {n(n+1)}$.
For 1) note that (by going to a subsequence) we may suppose $u_n(c) \to u(c)$for some $c$ and $u_n \to u$ a.e.. Now $u_n(x)=u_n(c)+\int_c^{x}u_n'(t)dt=u_n(c)+\int_c^{x}[u_n'(c)+\int_c^{t}u_n''(s)ds]dt$. I hope this is enough of a hint!.