  We show that any $N$-dimensional linear subspace of $L^2(\mathbb{T})$ admits
an orthonormal system such that the $L^2$ norm of the square variation operator
$V^2$ is as small as possible. When applied to the span of the trigonometric
system, we obtain an orthonormal system of trigonometric polynomials with a
$V^2$ operator that is considerably smaller than the associated operator for
the trigonometric system itself.
