Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $f_k$ be a sequence of non-negative 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 $F=f^2$?

share|cite|improve this question
No, thanks, but non-negativity seems to be important. –  user21629 Feb 23 '12 at 12:25

1 Answer 1

up vote 4 down vote accepted

The answer is "yes", even for non non-negative $f_n$. 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 $(f-f_n)^2$ 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.

share|cite|improve this answer
The unnecessary assumption that $f_n \ge 0$ makes the problem harder. –  Bill Johnson Feb 23 '12 at 18:29
I fixed the final equation in the first paragraph (from $f_n$ to $f_n^2$). –  Matthew Daws Feb 23 '12 at 20:05
Thank you, Matt. –  Bill Johnson Feb 23 '12 at 20:18
In the first sentence, do you really mean "even for non negative?" –  Yemon Choi Feb 23 '12 at 21:28
Thanks, Yemon. I corrected. –  Bill Johnson Feb 24 '12 at 1:06

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.