W^*-derived sets of transfinite order of subspaces of dual Banach spaces

Details W^*-derived sets of transfinite order of subspaces of dual Banach spaces W^*-derived sets of transfinite order of subspaces of dual Banach spaces

1993-03-29

arxiv.org/abs/math/9303203v1

Dokl. Akad. Nauk Ukrain. SSR. Ser. A, 1987, no. 10, 9-12

Mathematics

Functional Analysis

Scientific paper

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: $$ \Gamma _{(\alpha )}=\bigcup _{\beta <\alpha }((\Gamma _{(\beta )})_{(1)}. $$

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