Let $\{\phi_n\}$ be an orthonormal base in $L^2[0,1]$ such that for every continuous functions $f$ on $[0,1]$ one may find a sequence of complex numbers $\{\lambda_n\}$ with $f(x)=\sum \lambda_n\phi_n(x)$ almost every where.
Q. Let $f$ be a continuous function. Does there exist a sequence of complex numbers $\{\mu_n\}$ such that $f(x)=\sum \mu_n\phi_n(x)$ for every $x\in [0,1]$?
No.
We can redefine $\{\phi_n\}$ so that $\phi_n(1/2)=0$ for all $n$, which doesn't affect its property as an orthonormal basis of $L^2[0,1]$. This means that $$ \sum \mu_n \phi_n(1/2) = 0 $$ for any sequence of complex number $\mu_n$, so it cannot be equal to any $f$ such that $f(1/2)\ne 0$.