Let $f_k$ be a sequence of nonnegative functions from $L_2(\Omega)$, where $\Omega$ is a bounded open set. Assume that $f_k\to f$ weakly in $L_2$ and strongly in $L_p$, $\forall p<2$. Assume also that $f_k^2\to F$ weakly in $L_1$. Does it imply that $F=f^2$?

$\begingroup$ No, thanks, but nonnegativity seems to be important. $\endgroup$ – user21629 Feb 23 '12 at 12:25
The answer is "yes", even for non nonnegative $f_n$. First assume $f=0$. Since $f_n\to 0$ in $L_1$, WLOG by passing to a subsequence $f_n\to 0$ a.e. and hence $f_n^2\to 0$ a.e. But $f_n^2$ converges weakly in $L_1$ hence is uniformly integrable, whence $\f_n^2\_1 \to 0$.
The general case follows from the special case once you verify that $(ff_n)^2$ is weakly convergent in $L_1$. Since $f_n^2$ converges weakly in $L_1$, it is enough to show that $f_n f$ converges weakly in $L_1$ to $f^2$, which in turn follows from the facts that $f_n$ converges weakly in $L_2$ to $f$ and $fg$ is in $L_2$ for every $g$ in $L_\infty$
Unfamiliar background can be found in standard text books; in particular, the book of Albiac and Kalton.

$\begingroup$ The unnecessary assumption that $f_n \ge 0$ makes the problem harder. $\endgroup$ – Bill Johnson Feb 23 '12 at 18:29

$\begingroup$ I fixed the final equation in the first paragraph (from $f_n$ to $f_n^2$). $\endgroup$ – Matthew Daws Feb 23 '12 at 20:05

$\begingroup$ In the first sentence, do you really mean "even for non negative?" $\endgroup$ – Yemon Choi Feb 23 '12 at 21:28
