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 $\int_0^1f_i(x)g_i(x) dx\to 0$?
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5$\begingroup$ What about $f_{k}=e^{ikx}$ and $g_{k} = e^{-ikx}$? $\endgroup$– Paata IvanishviliCommented Jul 15, 2019 at 1:46
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$\begingroup$ Oops. You are right.Thank you. $\endgroup$– asvCommented Jul 15, 2019 at 1:49
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1 Answer
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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 $\int_{0}^{1}f_{k}(x)g_{k}(x)dx=1$.
answered Jul 15, 2019 at 1:56