Approximation of Subspaces of Symmetric Spaces Generated by Independent Functions

Let $E$ be a subspace of a symmetric space $X$ generated by $n$ independent identically distributed functions. It is proved that, under certain conditions on $X$, there exists a projection $P$, $\|P\|\le K$ ($K$ depending only on $X$) whose image contains $E$ and has dimension at most $Cn \ln(n + 1)$ ($C$ is independent of $n$ and $X$).