Mean Convergence of Vector--valued Walsh Series
  Given any Banach space $X$, let $L_2^X$ denote the Banach space of all measurable functions $f:[0,1]\to X$ for which   ||f||_2:=(int_0^1 ||f(t)||^2 dt)^{1/2}   is finite. We show that $X$ is a UMD--space (see \cite{BUR:1986}) if and only if   \lim_n||f-S_n(f)||_2=0 for all $f\in L_2^X$, where   S_n(f):=sum_{i=0}^{n-1} (f,w_i)w_i   is the $n$--th partial sum associated with the Walsh system $(w_i)$. 
Joerg Wenzel