On the complemented subspaces of the Schreier spaces (Q2717549)

The family of Schreier spaces \(\{X^{\alpha}\): \(\alpha<\omega_{1}\}\) where \(\omega_{1}\) is the first uncountable cardinal was defined by \textit{D. G. Alspach} and \textit{S. A. Argyros} [``Complexity of weakly null sequences'', Diss. Math. 321, (1992; Zbl 0787.46009)] as a generalization of a peculiar Banach space due to \textit{J. Schreier} [Stud. Math. 2, 58-62 (1930; JFM 56.0932.02)] which in this hierarchy is denoted by \(X^{1}\). It was proved there that the natural basis \(\{e_{n}^{\alpha}:\) \(n<\omega\}\) of the space \(X^{\alpha}\) is unconditional and shrinking for every \(\alpha<\omega_{1}\). NEWLINENEWLINENEWLINEIn the reviewed paper properties of subsequences of bases \(\{e_{n}^{\alpha}:\) \(n<\omega\}\) are studied. In particular, it is shown that for every finite ordinal \(\xi\) there exists a family \(A^{\xi}\) of infinite subsets of integers of cardinality equal to continuum which has the following property: for every distinct pair \(L,M\) of elements of \(A^{\xi}\) spaces \(X_{L}^{\xi}= \text{span}\{e_{n}^{\alpha}: n\in L\}\) and \(X_{M}^{\xi}=\) span\(\{e_{n}^{\alpha}: n\in M\}\) are of incomparable linear dimension.