I am looking for a reference for the following fact. Any hints would be appreciated.
Suppose $(x_n), (y_n)\subset [0,1]$ are some sequences, $(a_n)$ is absolutely summable and for each $f\in C[0,1]$ we have
$\sum_{k=1}^\infty a_k f(x_k) = \sum_{k=1}^\infty a_k f(y_k)$.
Then
$x_k = y_k$ for $k\in \mathbb{N}$.