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About the Banach envelope of \(l_{1,\infty}\). (English) Zbl 1175.46004
The author studies the Banach envelope \(l_{1,\infty}^{\text{ban}}\) of the quasi-Banach space \(l_{1,\infty}\) consisting of all sequences \(x = (\xi_k)\) with \(s_n(x) = O(\frac 1 n)\), where \((s_n(x))\) denotes the non-increasing rearrangement of \(x = (\xi _k)\). The situation turns out to be much more complicated than that in the well-known case of the separable subspace \(l_{1,\infty}^\circ\), whose members are characterized by \(s_n(x)=o(\frac 1 n)\). Namely, the Banach envelope of the latter space is known to be \(m_{1,\infty}^\circ\), the closed hull of all sequences with only a finite number of nonzero elements in the Sargent space \(m_{1,\infty}\) which is the collection of all \(x=(\xi_k)\) for which \(\| x | m_{1,\infty} \|:= \sup_n \frac{s_1(x)+ \ldots +s_n(x)}{\frac{1}{1} + \ldots +\frac{1}{n}}\) is finite. However, as the main result the author proves that the norms \(\| \cdot | m_{1,\infty} \|\) and \(\| \cdot | {l_{1,\infty}^{\text{ban}}} \|\) fail to be equivalent on \(l_{1,\infty}\). The proof uses an explicit formula for the norm \(\| \cdot | {l_{1,\infty}^{\text{ban}}} \|\) induced on \(l_{1,\infty}\) given in [N. J. Kalton and F. A. Sukochev, J. Reine Angew. Math. 621, 81–121 (2008; Zbl 1152.47014)]. For the convenience of the reader, the author provides an elementary proof of the inequality needed for the proof of the main result.

46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
46B45 Banach sequence spaces
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