# Weak convergence in $L^p$

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

• What about $f_{k}=e^{ikx}$ and $g_{k} = e^{-ikx}$? Jul 15, 2019 at 1:46
• Oops. You are right.Thank you.
– makt
Jul 15, 2019 at 1:49

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 $$\int_{0}^{1}f_{k}(x)g_{k}(x)dx=1$$.