I have a question :
Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such that $\sum_{n}\psi_n(t)\phi_n(x,t) = 0$, then can we imply $\psi_n(t) = 0$ for all $n$ ?. If not, can we give a counter example ?
In fact, I would like to change the question more complicatedly. That is
Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ and $x_0$ such that $\sum_{n}\psi_n(t)\phi_n(x_0,t) = 0$, then can we imply
$\psi_n(t) = 0$ for all $n$ ?. If not, can we give a counter example ?