  It is an English translation of the paper originally published in Russian and
Ukrainian in 1987. In the appendix of his book S.Banach introduced the
following definition Let $X$ be a Banach space and $\Gamma$ be a subspace of
the dual space $X^*$. The set of all limits of $w^{*}$-convergent sequences in
$\Gamma $ is called the $w^*${\it -derived set} of $\Gamma $ and is denoted by
$\Gamma _{(1)}$. For an ordinal $\alpha$ the $w^{*}$-{\it derived set of order}
$\alpha $ is defined inductively by the equality: $$ \Gamma _{(\alpha
)}=\bigcup _{\beta <\alpha }((\Gamma _{(\beta )})_{(1)}. $$
