Weak convergence in $L^p$

Question

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 $\int_0^1f_i(x)g_i(x) dx\to 0$?

Answer 1

The answer is No.

Take $f_{k}(x)=e^{ikx}$, and $g_{k}(x)=e^{-ikx}$. Then by Riemann--Lebesgue 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 $\int_{0}^{1}f_{k}(x)g_{k}(x)dx=1$.