A positive answer to the basis problem

lcub;x _n rcub; with lcub;x _n ,x* _n rcub; biorthogonal is a “uniformly minimal basis with quasifixed brackets and permutations” of a Banach spaceX if lcub;x _n rcub; andx* _n rcub; are both bounded. Moreover, there is an increasing sequence lcub;q _m rcub; of positive integers such that, for eachx′ ofX, settingq′(0)=0, $$x' = \sum\limits_{m = 0}^\infty { \sum\limits_{n = q'(m) + 1}^{q'(m + 1)} {x_{\pi '(n)}^ * (x')x_{\pi '(n)} ,} } $$ , where, for eachm≥1,q(m)+1≤q′(m)≤q(m+1) while $$\left\{ {\pi '(n)} \right\}_{n = q(m) + 1}^{q(m + 1)} is a permutation of \left\{ n \right\}_{n = q(m) + 1}^{q(m + 1)} .$$ . Then, for each subspaceY of a separable Banach spaceX, there exists a uniformly minimal basis with quasi-fixed brackets and permutations ofY, which can be extended to a uniformly minimal basis with quasi-fixed brackets and permutations ofX.