 Signed interval projection norm

Relative to a Schauder basis, take the supremum over successive finite intervals whose minima belong to a specified hereditary family $\mathcal{F}$, and signs ${\epsilon }_i\in \{-1,1\}$. If this supremum is bounded above by a multiple of the original norm and the family contains all singletons, it gives an equivalent norm. Composing two admitted signed interval projections costs at most a factor of two in this norm: split their nonempty intersections according to which interval supplies the later left endpoint. In each of the two resulting families the minima form a subset of one original list, so heredity preserves admissibility. This estimate lets bounded distortion of a Banach space give a uniform bound independent of the complexity of $\mathcal{F}$.