On a Property of the Franklin System in $C[0,1]$ and $L^1[0,1]$

A problem posed by J. R. Holub is solved. In particular, it is proved that if $\{\widetilde f_n\}$ is the normalized Franklin system in $L^1[0,1]$, $\{a_n\}$ is a monotone sequence converging to zero, and $\sup_{n\in\mathbb N}\|{\sum_{k=0}^na_k\widetilde f_k}\|_1<+\infty$, then the series $\sum_{n=0}^{\infty}a_n\widetilde f_n$ converges in $L^1[0,1]$. A similar result is also obtained for $C[0,1]$.