= Signed interval projection norm
{title2=$q_{\mathcal F}(x)=\sup\|\sum_i\varepsilon_iP_{E_i}x\|$}
Relative to a <Schauder basis>, take the supremum over successive finite intervals whose minima belong to a specified hereditary family $\mathcal F$, and signs $\varepsilon_i\in\{-1,1\}$. 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 $\mathcal F$.
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