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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.

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