Dilworth, S. J.; Ferenczi, V.; Kutzarova, Denka; Odell, E. On strongly asymptotic \(\ell_{p}\) spaces and minimality. (English) Zbl 1139.46013 J. Lond. Math. Soc., II. Ser. 75, No. 2, 409-419 (2007). First of all, recall that an infinite-dimensional Banach space \(X\) is minimal (resp., complementably minimal) if every infinite-dimensional subspace has a subspace (resp., complemented subspace) isomorphic to \(X\). Moreover, for a fixed \(1 \leq p \leq \infty\), a Banach space \(X\) with a basis \((e_i)\) is asymptotic \(l_p\) if there exists \(C < \infty\) and an increasing function \(f: \mathbb N \to \mathbb N\) such that, for all \(n \in \mathbb N\), every normalized block basis \((x_i)_{i=1}^{n}\) of \((e_i)_{i=f(n)}^{\infty}\) is C-equivalent to the unit vector basis of \(l_{p}^{n}\). As the authors themselves mention in the introduction of the paper under review, their main results can be summarized in the following slightly weaker form: Let \(1 \leq p \leq \infty\) and let \(X\) be a Banach space with a semi-normalized strongly asymptotic \(l_p\) basis \((e_i)\); then:(a) if \(X\) is minimal and \(1 \leq p <2\), then \(X\) is isomorphic to a subspace of \(l_p\), (b) if \(X\) is minimal and \(2 \leq p < \infty\), or if \(X\) is complementably minimal and \(1 \leq p \leq \infty\), then \((e_i)\) is equivalent to the unit vector basis of \(l_p\) (or \(c_0\) if \(p= \infty\)). They present some consequences about the number of non-isomorphic subspaces of a Banach space. Furthermore, for \(1\leq p <2\), they give examples of strongly asymptotic \(l_p\) basic sequences in \(l_p\) spanning minimal spaces that are not isomorphic to \(l_p\). Finally, they also pose some questions. Reviewer: Elói M. Galego (Sao Paulo) Cited in 8 Documents MSC: 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 46B03 Isomorphic theory (including renorming) of Banach spaces Keywords:Asymptotic \(l_p\) spaces; strongly asymptotic \(l_p\) spaces; minimality in Banach spaces PDFBibTeX XMLCite \textit{S. J. Dilworth} et al., J. Lond. Math. Soc., II. Ser. 75, No. 2, 409--419 (2007; Zbl 1139.46013) Full Text: DOI arXiv