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A property of convex basic sequences in \(L_1\). (English) Zbl 1095.46006

Let \((x_i)_{n=1}^\infty\) be a basic sequence in a Banach space \(X\) which is quasi-normalized, that is, \(0<\inf_n\| x_n\| \leq \sup_n\| x_n\| <\infty\). For \(k\in \mathbb N\), set \[ \gamma_k=\inf \left\| \sum\limits_{i=1}^m a_ix_i\right\| , \] where the infimum is taken over all integers \(m\geq k\) and all collections of scalars \(a_i\) for which there are numbers \(i_1<\ldots <i_k\leq m\) such that \(| a_{i_j}| \geq 1\) for every \(j=1,\ldots ,k\). The basic sequence is called convex if \(\sup_k\gamma_k=\infty\). The author proves that \(X\) is super-reflexive if and only if every quasi-normalized basic sequence in \(X\) is convex. The main result asserts that every subsequence of a normalized convex basic sequence in \(L_1\) which spans a complemented subspace contains a further subsequence equivalent to the unit vector basis of \(l_1\). Some problems are formulated.

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B25 Classical Banach spaces in the general theory
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