$\newcommand{\span}{\operatorname{span}}$Let $(X,\mu)$ be a standard probability space. Let $A\subset L^\infty(X,\mu)$ be a total subset of $L^2(X,\mu)$. Let $f\in L^\infty(X,\mu)$ be an arbitrary bounded function. Is it always possible to find a sequence $a_n\in\span(A)$ such that $||a_n-f||_2\rightarrow 0$ AND $||a_n||_{\infty} \leq ||f||_{\infty}$ ?
I have tried to start with an arbitrary sequence $b_n\in\span(A)$ converging to $f$ in the $L^2$ norm and tried to recover the desired result however I failed. It is not a problem to consider $L^\infty(X,\mu)$ as a von Neumann algebra to solve this problem. Any solution is welcome.