  The paper is devoted to two particular cases of the following general
problem. Let $\alpha$ and $\beta$ be two types of bases in Banach spaces. Let a
Banach space $X$ has bases of both types and a subspace $M\subset X^*$ contains
the sequence of biorthogonal functionals of some $\alpha$-basis in $X$. Does
$M$ contain a sequence of biorthogonal functionals of some $\beta$-basis in
$X$?
  The following particular cases are considered:
  $(\alpha, \beta)$=(Schauder bases, unconditional bases),
  $(\alpha, \beta)$=(Nonlinear operational bases, linear operational bases).
  The paper contains an investigation of some of the spaces constructed by
S.Belle\-not in ``The $J$-sum of Banach spaces'', J. Funct. Anal. {\bf 48}
(1982), 95--106. (These spaces are used in some examples.)
