Every separable Banach space has a basis with uniformly controlled permutations
Paolo Terenzi
GDML_Books, (2006), p.

There exists a universal control sequence p̅(m)m=1∞ of increasing positive integers such that: Every infinite-dimensional separable Banach space X has a biorthogonal system xₙ,xₙ* with ||xₙ|| = 1 and ||xₙ*|| < K for each n such that, for each x ∈ X, x=∑n=1∞xπ(n)*(x)xπ(n) where π(n) is a permutation of n which depends on x but is uniformly controlled by p̅(m)m=1∞, that is, nn=1m⊆π(n)n=1p̅(m)⊆nn=1p̅(m+1) for each m.

EUDML-ID : urn:eudml:doc:286047 
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm439-0-1,
     author = {Paolo Terenzi},
     title = {Every separable Banach space has a basis with uniformly controlled permutations},
     series = {GDML\_Books},
     year = {2006},
     zbl = {1121.46014},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm439-0-1}
}

Paolo Terenzi. Every separable Banach space has a basis with uniformly controlled permutations. GDML_Books (2006),  http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm439-0-1/