My question is probably very basic, sorry about that.


Let $\{f\_i\},\{g\_i\}$ be two sequences converging to 0 **weakly** in $L^p[0,1]$ for **any** $p<\infty$. Can one conclude that $\int\_0^1f\_i(x)g\_i(x) dx\to 0$?



Weak convergence in $L^p$
The answer is No.


Take $f\_{k}(x)=e^{ikx}$, and $g\_{k}(x)=e^{-ikx}$. Then by [Riemann--Lebesgue lemma](https://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma) we have $\lim\_{k \to \infty} \int\_{0}^{1}f\_{k}(x)h(x)dx = \lim\_{k \to \infty}\int\_{0}^{1} g\_{k}(x)h(x)=0$ for any $h \in L^{q}([0,1])$, for any $1\leq q\leq \infty$. On the other hand $\int\_{0}^{1}f\_{k}(x)g\_{k}(x)dx=1$. 


