Orthonormal Basis for $L^2(E)$

Question

Suppose $\{\varphi_n\}$ is orthonormal basis of $L^2(E)$. Then for every $f\in L^2(E)$ we have \begin{equation*} f=\sum_{n=1}^{\infty}\langle f,\varphi_n\rangle\varphi_n, \end{equation*} with this series absolutely convergent, so $\left\vert\langle f,\varphi_n\rangle\right\vert=0$ as $n\to\infty.$ Then \begin{equation*} \lim_{n\to\infty}\int_E f(x)\overline{\varphi_n(x)}dx=0. \end{equation*} Does this imply \begin{equation*} \lim_{n\to\infty}\int_E f(x)\varphi_n(x)dx=0? \end{equation*}

Answer 1

Yes, apply the previous line to $\overline {f}$ and the take the conjugate.