Abstract. The orthonormal Franklin spline system on ${\msbm R}$ is treated as a system of functions in the metric space $L^p({\msbm R})$, $0< p\le1$. It is proved that a given Franklin series converges unconditionally in this metric space if and only if the corresponding square function is in $L^p({\msbm R})$. Moreover, the latter takes place if and only if the maximal function for the partial sums is in $L^p({\msbm R})$.

AMS Subject Classification (1991): 42C10

Received February 18, 1999. (Registered under 2729/2009.)