  We show that D. L\'{e}pingle's $L_1(\ell_2)$-inequality \begin{equation*}
  \Big\| \big( \sum_n \mathbb E[f_n | \mathscr F_{n-1}]^2
  \big)^{1/2}\Big\|_1 \leq 2\cdot \Big\| \big( \sum_n f_n^2
  \big)^{1/2} \Big\|_1, \qquad f_n\in\mathscr F_n,
  \end{equation*} extends to the case where we substitute the conditional
expectation operators with orthogonal projection operators onto spline spaces
and where we can allow that $f_n$ is contained in a suitable spline space
$\mathscr S(\mathscr F_n)$. This is done provided the filtration $(\mathscr
F_n)$ satisfies a certain regularity condition depending on the degree of
smoothness of the functions contained in $\mathscr S(\mathscr F_n)$. As a
by-product, we also obtain a spline version of $H_1$-$BMO$ duality under this
assumption.

Functional Analysis;Probability